1

GOYAL B

ROTHERS PRAKASHAN

SummATivE ASSESSmENT

MULTIPLE CHOICE QUESTIONS [1 Mark]

A. important Questions

5. ArITHMETIC PrOgrESSIONSIMPOrTANT TErMS, DEFINITIONS AND rESULTS

• Somenumbersarrangedinadefiniteorder,accordingto a definite rule, are said to forma sequence.

• A sequence is called an arithmetic progression (AP), if thedifferenceof anyof its termsand thepreceding term is always the same.

i.e., tn + 1 – tn = constant. • The constant number is called the common

difference of the A.P. • Ifa is thefirst termandd thecommondifference

of an AP, then the general form of the AP is a, a + d, a + 2d, ...

• Let a be the first term and d be the commondifference of anAP, then, its nth term or generalis givenby

tn = a + (n – 1) d • If l is the last termof theAP, thennth term from

theend is thenth termofanAP,whosefirst termis l and commondifference is – d.

\ nth term from the end=Last term + (n – 1) (– d) ⇒ nth term from the end= l – (n – 1) d • Ifa, b, c, are inAP, then (i) (a + k), (b + k), (c + k) are inAP. (ii) (a – k), (b – k), (c – k) are inAP. (iii) ak,bk,ck, are inAP.

(iv) , , are in AP ( 0)a b c

kk k k

≠

• Remember the following while working withconsecutive terms in anAP.

(i) Three consecutive terms in an AP. a – d,a,a + d First term=a – d, commondifference=d Their sum=a – d + a + a + d = 3a (ii) Four consecutive terms in an AP. a – 3d, a – d, a + d, a + 3d First term :a – 3d, commondifference=2d Their sum= a – 3d + a – d + a + d + a

+ 3d = 4a

(iii) Five consecutive terms in an AP. a – 2d,a – d,a,a + d,a + 2d First term=a – 2d, commondifference=d • The sum Sn up to n terms of anAP whose first

term is a and common difference d is given by

[2 ( 1) ]

2n

nS a n d= + −

• If the first term and the last term of anAP are t1 and tn, then

1( ) (first term +last term)

2 2n n

n nS t t= + =

If t1 = a, the first term and tn = l, the last term,

then ( )2n

nS a l= +

• Sn – Sn – 1 = tn

1. The common difference of theA.P. 3, 1, –1, –3... is :

(a) –2 (b) 2 (c) –1 (d) 3 2. Thegeneral formof anA.P. is : (a) a, a – d,a – 2d, a – 3d, ...

(b) a, a + d,a + 2d, a + 3d, ...

(c) a, 2d, 3d, 4d, ... (d) noneof these 3. ThecommondifferenceoftheA.P.8,11,14,17,

20, ... is : (a) 2 (b) –2 (c) 3 (d) –3

4. An A.P. whose first term is 10 and commondifference is 3, is :

(a) 10, 13, 16, 19, ... (b) 5, 7, 9, 11, ... (c) 8, 12, 16, 20, ... (d) all of these 5. The commondifferenceof theA.P. –5, –1, 3, 7,

... is: (a) 2 (b) 3 (c) –4 (d) 4 6. Whichofthefollowinglistofnumbersdoesform

anA.P.?

(a) 2, 4, 8, 16, ... (b) 5 7

2, ,3 .....2 2

Assignments in Mathematics Class X (Term II)

2

GOYAL B

ROTHERS PRAKASHAN

(c) 0.22, 0.22, 0.222, 0.2222, ... (d) 1, 3, 9, 27, ... 7. The10th termof theA.P. 5, 8, 11, 14, .... is : (a) 32 (b)35 (c) 38 (d) 185 8. The list of numbers – 10, – 6, – 2, 2 .... is : (a) anA.P.witha = – 16 (b) anA.P.witha = – 4 (c) anA.P.witha = 4 (d) not anA.P. 9. InanA.P.,ifa=–7.2,d=3.6andtn=7.2,then

n is : (a) 1 (b)3 (c) 4 (d) 5 10. In anA.P., if tn = 4,d = – 4, andn = 7, thena

is : (a) 6 (b) 7 (c) 20 (d) 28 11. InanA.P., ifa=8.5,d=0,n=57, then tnwill

be : (a) 0 (b) 8.5 (c) 103.5 (d)150 12. Thenth termof theA.P. 2, 5, 8, ... is : (a) 3n – 1 (b) 2n – 1 (c) 3n – 2 (d) 2n – 3 13. IffirsttermofanA.P.is2andcommondifference

is – 2, then t7 is : (a) – 8 (b) – 10 (c) – 5 (d) 10

14. The11th termof theA.P. 5 55, , 0, ,...

2 2− − is :

(a) – 20 (b) 20 (c) – 30 (d) 30 15. Which termof theA.P. 4, 9, 14, 19, ... is 109? (a) 14th (b) 18th (c) 22nd (d) 16th 16. HowmanytermsarethereintheA.P.1,3,5,....

73, 75? (a) 28 (b) 30 (c) 36 (d) 38 17. If thenumbersa,b,c are inA.P., then : (a) b – a = c – b (b) b + a = c + b (c) a – b = b – c (d) noneof these 18. Howmany termsare there in theA.P.7,10,13,

..., 151? (a) 50 (b) 55 (c) 45 (d) 49 19. Which termof theA.P. 72, 63, 54, .... is 0? (a) 8th (b) 9th (c) 10th (d) 11th 20. Thefamousmathematicianassociatedwithfinding

the sumoffirst 100natural numbers is : (a) Pythagoras (b) Newton (c) Gauss (d) Euclid 21. Thesumoffirst16termsof theA.P.10,6,2, ...

is : (a) – 320 (b) 320 (c) – 352 (d)– 400

22. The sumoffirst 5multiples of 3 is : (a) 45 (b) 55 (c) 65 (d) 75 23. Thefirst termofanA.P. is–5and thecommon

difference is 2. The sum of first 6 terms of thisA.P. is:

(a) 0 (b) 5 (c) 6 (d) 15 24. In anA.P., ifa = 1, tn = 20 and Sn = 399, then

n is : (a) 19 (b) 21 (c) 38 (d) 42 25. The sumoffirstn natural numbers is : (a) n2 (b) n(n + 1)

(c)( 1)

2

n n + (d)

( 1)

2

n n −

26. If 4th termof anA.P. is 14 and its 12th term is70,what is itsfirst term?

(a) – 10 (b) – 7 (c) 7 (d) 10 27. The first term of an A.P. is 6 and its common

difference is 5.Whatwill be its 11th term? (a) 56 (b) 41 (c) 46 (d) 50 28. The first four terms of anA.P. whose first term

is – 2 and the commondifference is – 2, are : (a) – 2, 0, 2, 4 (b) – 2, 4, – 8, – 16 (c) – 2, – 4, – 6, – 8 (d) – 2, – 4, – 8, – 16 29. If the common difference of anA.P. is 5, then

what is t18 – t13? (a) 5 (b) 20 (c) 25 (d) 30 30. ThefirsttwotermsofanA.P.are–3and4.The

21st termof theA.P.will be : (a) 17 (b) 137 (c)143 (d)– 143 31. TwoA.P.shavethesamecommondifference.The

first term of one of these is – 2 and that of theother is – 10.The difference between their 10thterms is :

(a) 8 (b) – 8 (c) 4 (d) – 4 32. If 2nd termof anA.P. is 13 and the 5th term is

25, then its 7th term is : (a) 30 (b) 33 (c) 37 (d) 38 33. Howmanynumbersof2-digitsaredivisibleby3? (a) 10 (b) 20 (c) 30 (d) 40 34. If11timesthe11thtermofanA.P.isequalto7

times its 7th term, then its 18th termwill be : (a) 7 (b) 11 (c) 18 (d) 0 35. What is the sum of all natural numbers from 1

to 100? (a) 4550 (b) 4780 (c) 5050 (d) 5150 36. 51+52+53+ ... + 100= ? (a) 3775 (b) 4025 (c) 4275 (d) 5050

3

GOYAL B

ROTHERS PRAKASHAN

37. Ifthesumofp termsofanA.P. isqandthesumof q terms is p, then the sum of (p + q) termswill be:

(a) 0 (b) p – q (c) p + q (d) – (p + q) 38. If the sumofn termsof anA.P. be3n2 – n and

its common difference is 6, then its first termis:

(a) 2 (b) 3 (c) 1 (d) 4 39. If the sum of n terms of anA.P. be 3n2 + 5n,

thenwhichof its terms is 164? (a) 26th (b)27th (c) 28th (d)noneof these 40. IfthesumofntermsofanA.P.is2n2+5n,then

itsnth term is :

(a) 4n – 3 (b) 3n – 4 (c) 4n + 3 (d) 3n + 4 41. If the first term of an A.P. is 2 and common

difference is 4, then the sumof its 40 terms is: (a) 3200 (b) 1600 (c) 200 (d) 2800 42. 25+28+31+ .... + 100= ? (a) 1625 (b) 1525 (c) 1725 (d)1650 43. The sumoffirstn positive integers is givenby:

(a) ( 1)

2

n n − (b) (2 1)

2

n n +

(c) ( 1)

2

n n + (d) noneof these

44. The sumoffirstn even integers is givenby :

(a) n(n + 1) (b) ( 1)

2

n n +

(c) (n + 1)(n – 1) (d) noneof these

B. Questions From CBSE Examination Papers

1. Which term of theA.P. 24, 21, 18, ...................is thefirst negative term? [2011 (T-II)]

(a) 8th (b) 9th (c) 10th (d) 12th 2. Which termof theA.P.2,–1,–4, .......... is–70? [2011 (T-II)] (a) 15th (b) 18th (c) 25th (d) 30th 3. The 4th term from the end ofA.P. –11, –8, –5,

................... 49 is : [2011 (T-II)]

(a) 37 (b) 40 (c) 43 (d) 58 4. Which termof theA.P. 21, 42,, 63 ...................

is 210? [2011 (T-II)] (a) 9th (b) 10th (c) 12th (d) 11th 5. What is the common difference of the A.P. in

whicha18 – a14 = 32 ? [2011 (T-II)]

(a) 8 (b) –8 (c) 4 (d) –4

6. Thenext termof theA.P. 27, 48, 75 , .... is : [2011 (T-II)]

(a) 105 (b) 107 (c) 108 (d) 147 7. Which termof theA.P. 100, 90, 80....... is zero?

[2011 (T-II)] (a) 5th (b) 6th (c) 10th (d) 11th 8. Whichof the following is not anA.P.

[2011 (T-II)] (a) 13, 8, 3, –2, –7, –12 (b) 10.8, 11.2, 11.6, 12, 12.4

(c)1 2 3 4 5

8 ,18 ,28 ,48 ,587 7 7 7 7

(d) 3 6 9 128 ,11 ,14 ,17

23 23 23 23

9. Which termof theA.P. 92, 88, 84, 80, .... is 0?[2011 (T-II)]

(a) 23 (b) 32 (c) 22 (d) 24 10. For anA.P. ifa25 – a20 = 45, thend equals to :

[2011 (T-II)]

(a) 9 (b) –9 (c) 18 (d) 23

11. If the nth term of anA.P. is 3

4

n+ , then its 8thterm is : [2011 (T-II)]

(a) 11 (b) 11

4 (c) 11

2 (d) 22

12. Ifa,a – 2 and 3a are inA.P., then thevalueofa is : [2011 (T-II)]

(a) –3 (b) –2 (c) 3 (d) 2 13. If anA.P.hasa=1, tn = 20 and Sn=399, then

valueofn is : [2011 (T-II)]

(a) 20 (b) 32 (c) 38 (d) 40 14. If a + 1, 2a + 1, 4a – 1 are in A.P., then the

valueofa is : [2011 (T-II)]

(a) 1 (b) 2 (c) 3 (d) 4 15. ThesumofthefirstntermsofanA.P.is2n2+5n.

Then itsnth term is : [2011 (T-II)]

(a) 4n + 3 (b) 4n – 3 (c) 3n – 4 (d) 3n + 4 16. Forwhatvalueofp are 2p+1,13,5p –3,three

consecutive termsof anA.P.? [2011 (T-II)]

(a) 2 (b) –2 (c) 4 (d) –4 17. Whichtermof theA.P.,113,108,103, ...............

is thefirst negative term? [2011 (T-II)]

(a) 22nd term (b) 24th term (c) 26th term (d) 28th term

4

GOYAL B

ROTHERS PRAKASHAN

18. 15th termof theA.P.,x –7,x –2,x+3,....is :[2011 (T-II)]

(a) x + 63 (b) x + 73

(c) x + 83 (d) x + 53 19. Ifp–1,p+3,3p–1areinA.P.,thenp isequal

to: [2011 (T-II)] (a) 4 (b) – 4 (c) 2 (d) – 2

SHOrT ANSWEr TYPE QUESTIONS [2 Marks

A. important Questions

1. FortheA.P.–3,–7,–11,...canwefinddirectlyt30 – t20withoutactuallyfindingt30 and t20?Givereasons for your answer.

2. Is 68 a termof theA.P. 7, 10, 13, ...?

3. The 4th term of anA.P. is three times the firstterm and the 7th term exceeds twice the thirdterm by 1. Find the first term and the commondifference.

4. Shonal deposited Rs 1000 at compound interestat the rate of 10% p.a. The amounts at the endof first year, second year, third year .... form anA.P. Justify your answer. [HOTS]

5. ThenthtermofanA.P.cannotbem2+1.Justifyyour answer.

6. Writetheexpressiontn – tmfortheA.P.a,a + d,a + 2d,.....Hence,findthecommondifferenceoftheA.P. forwhich 11th term is 5 and 13th termis 79.

7. Show that x – y, x and x + y form consecutivetermsof anA.P.

8. Justifywhetheritistruetosaythat2n–3isthenth termof anA.P.

9. Which term of the A.P. 32, 29, 26, ... is firstnegative term?

10. Is 0 a term of theA.P. 31, 28, 25, ... ? Justifyyour answer.

11. If the numbers of x – 2, 4x – 1, and 5x + 2 are inA.Pfind thevalueofx.

12. Verify that eachof the following is anA.P. : (a) x + y, (x + 1) + y, (x + 1) + (y + 1), ... (b) x, 2x + 1, 3x + 2, 4x + 3, ... 13. Find a,b and c, if thenumbersa,7,b,23,c are

inA.P.. 14. Find the number of terms of theA.P. 17, 14.5,

12, .... – 38. 15. The sum of n terms of a sequence is 3n2 + 4n.

Find thenth termand show that it is anA.P.. 16. The sum of first n terms of anA.P. is given by

3n2 – n.Determine theA.P. and its 25th term.

17. How many terms of theA.P. 116 , 5...

2− − − are

needed to give the sum–25? 18. Find the sumoffirstn oddnatural numbers. 19. Showthatthesumofalloddintegersbetween1and

100whicharedivisibleby3 is83667. [HOTS]

20. Iftn = 3 – 4n,showthatt1,t2,t3,...formanA.P.Also,findS20. [HOTS]

B. Questions From CBSE Examination Papers

1. Which termof theA.P., 6, 13, 20, 27, ...............is 98more than its 24th term? [2011 (T-II)]

2. Calculate how many multiples of 7 are therebetween100 and300. [2011 (T-II)]

3. IfSndenotesthesumofntermsofanAPwhosecommon difference isd and first term isa, findSn – 2Sn–1 + Sn–2. [2011 (T-II)]

4. Find the sum of all two digit positive numbersdivisible by3. [2011 (T-II)]

5. Determine the 2nd term of an A.P. whose 6thterm is 12 and8th term is 22. [2011 (T-II)]

6. Find the sum of first 10 terms of the sequence{an} where an = 5 – 6n, where n is a naturalnumber. [2011 (T-II)]

7. Find the sum of the first 50 odd naturalnumbers. [2011 (T-II)]

8. If the nth term of anA.P. is (2n + 1), find thesumoffirstn termsof theA.P. [2011 (T-II)]

9. The 8th term of anA.P. is 37 and its 12th termis 57.Find theA.P. [2011 (T-II)]

10. Which term of theA.P. 3, 15, 27, 39,............. is132more than its 54th term? [2011 (T-II)]

11. In anA.P. the first term is – 4, the last term is29 and the sum of all its terms is 150.Find thecommondifferenceof theA.P. [2011 (T-II)]

12. For anA.P. show that 2 2p p p qa a a+ ++ =

[2011 (T-II)]

13. FindthecommondifferenceofanA.P.whosefirst

term is 1

2andthe8thtermis 17

6.Alsowriteits

4th term. [2011 (T-II)]

5

GOYAL B

ROTHERS PRAKASHAN

23. If thesumoffirstn termsofanA.P. isgivenbySn = 4n2 – 3n, find thenth termof theA.P. [2004C]

24. If thesumoffirstn termsofanA.P. isgivenbySn = 2n2 + 5n, find thenth termof theA.P. [2004C]

25. Find thenumberof termsof theA.P.54,51,48,.... so that their sum is 513. [2005]

26. Find the sum of the first 51 terms of the A.P.whose2nd term is 2 and4th term is 8.[2005C]

27. The sumof the firstn terms of anA.P. is givenbySn = 3n2 – n.Determine theA.P.and its25thterm. [2005C]

28. ThesumofthreenumbersinA.P.is27andtheirproduct is 405.Find thenumbers. [2005C]

29. HowmanytermsarethereinanA.P.whosefirstterm and 6th term are – 12 and 8 respectivelyand sumof all its terms is 120? [2006]

30. In an A.P. the sumoffirstn terms is25 3

.2 2

n n+

Find its 20th term. [2006C] 31. The first term, common difference and last term

of anA.P. are 12, 6 and 252 respectively. Findthe sumof all termsof thisA.P. [2007]

14. Find the sum of first twelve multiples of 7. [2011 (T-II)]

15. Find the sumof the 25 termsof anA.P. whosenth term is givenby tn = 7 – 3n. [2011 (T-II)]

16. Howmany terms are there inA.P. 7, 16, 25, .........., 349? [2011 (T-II)]

17. Find thenumberof termsof the series: – 5+ (–8) + (–11)+ ............. + (– 230) [2011 (T-II)] 18. Which term of the arithmetic progression 3, 10,

17..........will be 84more than its 13th term? [2011 (T-II)]

19. Find thenumberof all2digitnumbersdivisibleby3. [2011 (T-II)]

20. Find the value of p, if the numbers x, 2x + p, 3x + 6, are three consecutive termsof anA.P. [2011 (T-II)]

21. If 6th term of anA.P. is –10 and its 10th termis –26, then find the 15th term of the A.P. [2011 (T-II)]

22. If 8th termof anA.P. is 31 and15th term is 16more than11th term,find theA.P. [2011 (T-II)]

SHOrT ANSWEr TYPE QUESTIONS [3 Marks]

A. important Questions

1. Determine theA.P.whose5th term is19and thedifferenceoftheeighttermfromthethirteethtermis20.

2. Thesumoffirst threetermsofanA.P. is21andtheir product is 231.Find thenumbers. [Imp.]

3. WhatarethemiddlemosttermsoftheA.P.–11,– 7, – 3, ... 49?

4. The 26th, 11th and the last term of anA.P. are

0, 3 and 1

5− respectively. Find the common

difference and thenumberof terms. 5. If the 9th term of anA.P. is zero, prove that its

29th term is twice its 19th term. 6. The sum of first 6 terms of anA.P. is 42. The

ratio of the 10th term to the 30th term is 1 : 3.Find thefirst termand the11th termof theA.P.

7. If the nth terms of two A.Ps. 9, 7, 5, ... and 24,21,18,...aresame,findthevalueofn.Also,find that term.

8. Howmanynumbersliebetween10and30whichwhendividedby4 leave a remainder 3?

9. If (m+1)th termofanA.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.

10. The sum of first three terms of anA.P. is 33. Iftheproductofthefirstandthethirdtermsexceedsthe second termby29,find theA.P.

11. Find the sum = 3 2 5 3...

a b a b a b

a b a b a b

+ − −+ + +

− + + to

11 terms. 12. IfSndenotes thesumoffirstn termsofanA.P.,

prove thatS12 = 3(S8 – S4). 13. If the numbers a, b, c, d, e, form anA.P., then

find thevalueofa – 4b + 6c – 4d + e. 14. In an A.P. if the first term is 22, the common

difference is – 4 and the sum of n terms is 64,findn.

15. Split 207 into three parts such that these are inA.P. and the product of the two smaller parts is4623.

16. Howmanytermsof theA.P.–15,–13,–11, ...are needed to make the sum – 55? Explain thereason for thedouble answer. [HOTS]

17. The sum of first n terms of anA.P. whose firstterm is 8 the common difference is 20 is equaltothesumoffirst2ntermsofanotherA.P.whoefirst term is – 30 and the common difference is8.Findn.

6

GOYAL B

ROTHERS PRAKASHAN

18. If a, b, c are in A.P., show that (a – c)2 = 4 (b2 – ac).

19. ReshmasavesRs32duringthefirstmonth,Rs36inthesecondmonthandRs40inthethirdmonth.If she continues to save in this manner, in howmanymonthswill she saveRs2000? [HOTS]

20. Amanufacturerof radio setsproduced800unitsinthethirdyearand700unitsintheseventhyear.Assumingthattheproductincreasesuniformlybyafixednumber everyyear,find

(i) theproduction in thefirst year (ii) theproduction in the10thyear.

21. The pth term of anA.P is q and qth term is p.Find its (p + q)th term.

22. Ifa,b,c are inA.P., show that

(i) 1 1 1

, , are in A.P.bc ca ab

(ii) a2 (b + c),b2(c + a),c2(a + b) are inA.P.

23. If a2, b2, c2 are in A.P., then prove that1 1 1

, , are in A.P.b c c a a b+ + +

[HOTS]

24. If the sum of m terms of an A.P. is the sameas the sum ofn terms, show that the sum of its (m + n) terms is 0. [HOTS]

B. Questions From CBSE Examination Papers

1. Which termof theA.P. : 3, 15, 27, 39, ......willbe 120more then its 21st term? [2011 (T-II)]

2. ThesumofthefirstntermsofanA.P.is5 32n n- . Find the A.P. and hence find its 12th term. [2011 (T-II)]

3. The angles of a triangle are inA.P.Thegreatestangle is twice the least. Find all angles of thetriangle. [2011 (T-II)]

4. Findthesumofallnaturalnumbersbetween200and1000 exactlydivisible by6. [2011 (T-II)]

5. Find thevalueof themiddlemost term(s)of thearithmetic progression : [2011 (T-II)]

– 11, – 7, – 3, ............49. 6. The sum of first six terms of an arithmetic

progressionis42.Theratioofits10thtermtoits30thtermis1:3.Findthefirstandthethirteenthtermof theA.P. [2011 (T-II)]

7. HowmanytermsoftheA.P.9,17,25, .....,mustbe taken to get a sumof 450 ? [2011 (T-II)]

8. Find the sum of all two digit odd positivenumbers. [2011 (T-II)]

9. If the sum of first fourteen terms, of an AP is1050 and itsfirst term is 10,find its 20th term. [2011 (T-II)]

10. If the sumoffirstn termsof anA.P. is4n2 – n,find the12th term. [2011 (T-II)]

11. The sum of first three terms of anA.P. is 33. Iftheproductofthefirstandthirdtermexceedsthesecond termby29,find theA.P. [2011 (T-II)]

12. Whichtermisthefirstnegativeterminthegiven

A.P. : 23 211

220, , , ........ ? [2011 (T-II)]

13. In anA.P. the first term is –4, the last term is29 and the sum of all its term is 150. Find itscommondifference. [2011 (T-II)]

14. TheticketreceiptsattheshowofafilmamountedtoRs6,500onthefirstdayandshowedadropofRs 110 every succeeding day. If the operationalexpensesoftheshowareRs1000aday,Findonwhichday the showceases to beprofitable. [2011 (T-II)]

15. Findthesumofallthetwo-digitnaturalnumberswhich are divisible by4. [2011 (T-II)]

16. Determine ‘a’ so that 2 1 12a a a+ + +, and 3 3 32a a- + are consecutive termsof anA.P. [2011 (T-II)]

17. If the sum of firstm terms of anA.P. is n and thesumoffirstn termsinm, thenshowthat thesumof itsfirst (m + n) terms is – (m + n). [2011 (T-II)]

18. Howmany terms of theA.P. – 6, –11/2, –5, .....are needed to give the sum–25 ? [2011 (T-II)]

19. If thesumofall thetermsofanA.P.1,4,7,10,.........,x. is 287,findx. [2011 (T-II)]

20. Howmany terms of theA.P. 78, 71, 64, ..... areneeded to give the sum 465 ?Also find the lasttermof thisA.P. [2011 (T-II)]

21. The 4th term of anA.P. is equal to 3 times thefirst termandthe7thtermexceedstwicethe3rdterm by 1. Find the first term and the commondifference. [2011 (T-II)]

22. Findthreenumbers inA.P.whosesumis15andwhoseproduct is 105. [2011 (T-II)]

23. The sum of the 5th and 7th terms of anA.P. is52 and its 10th term is 46.Find theA.P. [2011 (T-II)]

24. Sum of the first n terms of an A.P. is 5n2 – 3n.Find the A.P. and also find its 16th term. [2011 (T-II)]

25. Find the sum of all three digit numbers whichleave the same remainder 2whendividedby5. [2011 (T-II)]

7

GOYAL B

ROTHERS PRAKASHAN

26. ForwhatvalueofnarethenthtermsoftwoA.P.63, 65, 67 ... and3, 10, 17 ... equal ? [2008]

27. If the sum of first 7 terms of anA.P. is 49 andthatoffirst17termsis289,findthesumoffirstn terms. [2008C]

28. Ifthe8thtermofanA.P.is37andthe15thtermis15morethanthe12thterm,findtheA.P.Hencefind the sumof thefirst 15 termsof theA.P. [2008C]

29. The sumof first six terms of anA.P. is 42.Theratio of its 10th term to its 30th term is 1 : 3.CalculatethefirstandthethirteenthtermsoftheA.P. [2009]

30. The sum of the first sixteen terms of anA.P. is112andthesumofitsnextfourteentermsis518.Find theA.P. [2010]

B. Questions From CBSE Examination Papers

1. InanA.P.thesumoffirsttentermsis–80andthesumof next ten terms is –280.Find theA.P.

[2011 (T-II)] 2. Thesumoffirst7termsofanA.P.is49andthat

of first 17 terms is 289. Find the sum of first n terms. [2011 (T-II)]

3. Kartik repays his total loan of Rs. 1,18,000by paying every month starting with the first

instalment of 1000. He increases the instalmentby Rs 100 every month.What amount will bepaidbyhiminthe30thinstalment?Whatamountof loan does he still have to pay after the 30thinstalment? [2011 (T-II)]

4. In anA.P., the sum of first n terms is given by23 5

2 2n

n nS = + . Find the25th termof theA.P.

LONg ANSWEr TYPE QUESTIONS [4 Marks]

A. important Questions 1. Thesumof fourconsecutivenumbers inanA.P.

is32andtheratioof theproductof thefirstandthe last terms to the product of the two middleterms is 7 : 15.Find thenumbers.

2. If 1 1 1, ,

a b c are inA.P., show that :

(i) , ,b c c a a b

a b c

+ + + are inA.P.

(ii) bc,ca,ab are inA.P.

3. Find the sumof those integers : (a)between1and500whicharemultiplesof2as

well as of 5. (b)from1to500whicharemultiplesof2aswell

as of 5. (c)from 1 to 500 which are multiples of 2

or 5. 4. Ifpth,qth and rth termsof anA.P. area,b and

c respectively, then show that : (a) a(q – r) + b(r – p) + c(p – q) = 0 (b) (a – b)r + (b – c)p + (c – a)q = 0 5. Solve the equation 1+4+7+10+ .... +x = 287. 6. The sums of x terms of three A.P.s are S1, S2

and S3. The first term of each is unity and thecommon differences are 1, 2 and 3 respectively.Prove thatS1 + S3 = 2S2.

7. AnA.P. consists of 37 terms. The sum of threemiddlemosttermsis225andthesumofthelastthree is 429.Find theA.P.

8. IfmthtermofanA.P.is 1

nandthenthtermis 1

m,

show that the sumof itsmn terms is 1( 1)

2mn + .

9. If S1, S2, S3 be the sums of n, 2n and 3n termsrespectivelyofanA.P,prove thatS3 = 3(S2 – S1).

10. Prove that no matter what the real numbers a and b are, the sequencewith nth term a + nb is always anA.P.What is the common difference?What is the sumof thefirst 20 terms? [HOTS]

11. A contractor employed 150 labourers to finish apieceofworkinacertainnumberofdays.4workerswent away the second day, 4moreworkerswentaway the third day and so on. If it took 8 moredays to finish thework, find the number of daysinwhich theworkwascompleted. [HOTS]

12. The students of a school decided tobeautify theschoolontheannualdaybyfixingcolourfulflagson the straightpassageof the school.Theyhave27flags tobefixedat intervalofevery2m.Theflags are stored in the position of the middlemost flag.Ruchiwas given the responsibility ofplacingtheflags.Ruchikeptherbookswheretheflagswere stored. She could carry only oneflagat a time. Howmuch distance did she cover incompletingthisjobsandreturningbacktocollecther books? What is the maximum distance shetravelled carrying aflag? [HOTS]

8

GOYAL B

ROTHERS PRAKASHAN

5. Acontractoronconstructionjobspecifiesapenaltyfor delay of completion beyond a certain dateas follows :Rs200 for thefirst day,Rs250 forthesecondday,Rs300 for the thirddayetc. thepenaltyforeachsucceedingdaybeingRs50morethanfortheprecedingday.Howmuchmoneythecontractorhastopayaspenaltyifhehasdelayedtheworkby30days ? [2011 (T-II)]

6. Ifa,bandcbethesumsoffirstp,q and rtermsrespectivelyof anAP, show that

( ) ( ) ( ) 0

a b cq r r p p q

p q r− + − + − = [2011 (T-II)]

7. InNovember2009,thenumberofvisitorstoazooincreaseddailyby20.Ifatotalof12300peoplevisitedthezoointhatmonth,findthenumberofvisitors on1stNovember2009. [2011 (T-II)]

8. For what value of n, the nth terms of theA.P. 63, 65, 67, .... and 3, 10, 17, ......... are equal?Alsofind that term. [2011 (T-II)]

9. Aladderhasrungs25cmapart.Therungsdecreaseuniformly in length from 45 cm at the bottomto 25 cm at the top (see figure). If the top andbottomrungsare2.5mapart,what is the lengthof thewood required for the rungs? [2011 (T-II)]

10. Findthesumofthefirst31termsofanA.P.whose

nth term is givenby3+2

3 n. [2011 (T-II)]

11. ThesumofthethirdandseventhtermofanA.P.is 6 and their product is 8. Find the sum of thefirst sixteen termsof theA.P. [2011 (T-II)]

12. AsumofRs2700istobeusedtogiveeightcashprizes to students of a school for their overallacademic performance. If each prize is Rs 25more than its preceding prize find the value ofeachof theprizes. [2011 (T-II)]

13. InanA.P.,provethat 2 , wherem n m n m aa a a a+ -+ =

denotesnth termof theA.P. [2011 (T-II)]

14. A spiral is made up of successive semicircles,withcentresalternativelyatAandB,startingwithcentreatAofradii0.5cm,1cm,1.5cm,2cm....

as shown in the figure.what is the total length

of such spiral made up of thirteen consecutive

semicircles?22

(Take )7

π = [2011 (T-II)]

15. AmanufacturerofT.V.setsproduced600setsinthe third year and 700 sets in the seventh year.Assumingthattheproductionincreasesuniformlyby afixednumber everyyear,find

(a) Theproduction in thefirst year

(b) Theproduction in the10thyear

(c) The total production infirst 7 years [2011 (T-II)]

16. Thehousesofarowarenumberedconsecutivelyfrom1 to49.There isavalueofx such that thesumof thenumbers of thehousespreceding thehouse numbered x is equal to the sum of thenumberofthehousesfollowingit.Findthisvalueofx. [2011 (T-II)]

17. Showthat thesumofalloddintegersbetween1and1000which are divisible by3 is 83667.

[2011 (T-II)]

18. The 4th term of anA.P. is equal to 3 times thefirst termandthe7thtermexceedstwicethe3rdtermby1.Find theA.P. [2011 (T-II)]

19. Find thesumofallmultiplesof9 lyingbetween300 and700. [2011 (T-II)]

20. A woman takes up a job of Rs 8000 per monthwith an annual increment of Rs 100.What willsheearnover aperiodof10years ? [2011 (T-II)]

21. 228 logs are to be stacked in a store in thefollowing manner: 30 logs in the bottom, 28 inthe next row, then 26 and so on. In how manyrows can these228 logsbe stacekd?Howmanylogs are there in the last row? [2011 (T-II)]

22. NeerasavesRs1600duringthefirstyear,Rs2100in the second year, Rs 2600 in the third year. Ifshe continues her savings in this pattern, in howmanyyearswill she saveRs38500? [2011 (T-II)]

23. Ifm times themth termof anA.P. is equal ton timesitsnthterm,showthatthe(m+n)thtermoftheA.P. is zero (m ≠ n). [2011 (T-II)]

9

GOYAL B

ROTHERS PRAKASHAN

FORmATivE ASSESSmENT

Activity-1

Objective : To verify that the given sequence is anarithmetic progression by paper cutting and pastingmethod.

Materials requried : Squared paper, colour pencils,geometrybox,etc.

Procedure :

Case 1.Letthegivensequencebe1,4,7,10,13,...

1. Take a squared paper. Using a colour pencil,colour 1 square (1 is thefirst termof thegivensequence).

2. Using a different colour pencils, colour 4squaresvertically(4isthe2ndtermofthegivensequence).

3. Colour7squaresvertically(7isthe3rdtermofthesequence).

4. Colour10squaresvertically(10isthe4thtermofthesequence)andsoon.

Figure 2Figure 1

Case 2.Letthesequencebe1,3,6,8,10,12,…

1. Takeasquaredpaper.Usingacolourpencil,colour1square(1isthefirsttermofthesequence).

2. Usingadifferentcolourpencil,colour3squaresvertically(3isthesecondtermofthesequence).

3. Nextcolour6squaresvertically(6isthethirdtermofthesequence).

4. Colour8squaresvertically(8isthefourthtermofthesequence)andsoon.

Observations : 1. For figure 1: The difference between the heights of first and

secondstrips=3cm. Thedifferencebetweentheheightsofsecondand

thirdstrips=3cm. Thedifferencebetween theheightsof third and

fourthstrips=3cm. Thedifferencebetweentheheightsoffourthand

thefifthstrips=3cm. 2. Ifweconsiderfigure1asaladder,thenwecansay

thatthedifferencebetweentheheightsofadjoiningstepsofthisladderisconstant,whichis3cm.

3. For figure 2 : The difference between the heights of first and

secondstrips=2cm. Thedifferencebetweentheheightsofsecondand

thirdstrips=3cm. Thedifferencebetween theheightsof third and

fourthstrips=2cm. Thedifferencebetweentheheightsoffourthand

fifthstrips=2cm. 4. Ifweconsiderfigure2asaladder,thenwecansay

thatthedifferencebetweentheheightsofadjoiningstepsofthisladderisnotconstant.

Conclusion : Squence 1 is anAP, as the differencebetweenitsconsecutivetermsisconstant.Sequence2isnotanAP,asthedifferencebetweenitsconsecutivetermsisnotconstant.

Activity-2

Objective : To verify that the sumof firstn natural

numbers is n n( )+1

2byactivitymethod.

Materials required : Squared paper, colour pencils,geometybox,apairofscissors,whitesheetsofpaper,gluestick,etc.

Procedure : Let us find the sum of first 12 naturalnumbers,ie,1+2+3+4+…+12. 1. Takeasquaredpaperofsize12cm×12cm.With

thehelpofcolourpencils,shaderectanglesoflength1cm,2cm,3cm,4cm,…12cmandofwidth1cmeach,asshowninfigure3.Marktherectanglesas1,2,3,…12respectivetyfromtoptobottom.

10

GOYAL B

ROTHERS PRAKASHANFigure 3

Figure 4

2. Takeanothersquaredpaperofsize12cm×12cm.Shaderectanglesoflength1cm,2cm,3cm,…12cmandofwidth1cmeach,inriverseorderasshowninfigure4.Marktherectanglesas1,2,3,…12respectivetyfrombottomtotop.

3. Now,cutoutthecolouredportionsoffigures3and4andpastethemtogetheronawhitesheetofpaperasshowninfigure5.

Figure 5

Observations : 1. Theareaoftheshadedregioninfigure3 = (1×1+1×2+1×3+1×4+…+1×12)cm2

=(1+2+3+4+…+12)cm2

2. Theareaoftheshadedregioninfigure4

=(1×1+1×2+1×3+1×4+…+1×12)cm2 = (1 + 2+3+4+…+12)cm2

3. Since,figure5iscomprisedofshadedregionsoffigures1and2so,areaoffigure5 =2(1+2+3+4+…+12)cm2

4. But,figure5isarectangleofdimensions12cm×13cmor12cm×(12+1)cm

So,itsarea=12(12+1)cm2

5. From 3and4above,2(1+2+3+4+…+12)=12×(12+1)

⇒ 1+2+3+4+…+12=12 (12 1)

2

× + .

⇒sumoffirst12naturalnumbers=12(12 1)

2

+.

So,sumoffirstnnaturalnumbers=1+2+3+…

+ n = ( 1)

2

n n +.

Do Yourself : Repeattheaboveactivitybyconsideringthesumoffirst20naturalnumbers.

Activity-3

Objective :Toverifythatthesumoffirstnoddnaturalnumbersisgivenbyn2byanactivitymethod.

Materials required : Squared paper, colour pencils,whitesheetsofpaper,apairofscissors,geometrybox,gluestick,etc.

Procedure :

1. Takeasquaredpaperofsize10cm×10cm.Usingacolourpencil,shadeitstopleftsquare.

2. Colour the three squares adjacent to the abovesquare,usingadifferentcolour.

3. Colourthefivesquares,whichareadjacenttotheprevioussquaresusingadifferentcolour.

4. Colour seven squareswhich are adjacent to theprevioussquares,usingadifferentcolour.

5. Continue this process till you shade all thesquares.

11

GOYAL B

ROTHERS PRAKASHANFigure 6

Observations : 1. Areaofthesquareshadedinstep1=1cm2

2. Totalareaofthesquaresshadedinsteps1and2 =(1+3)cm2

3. But,thesquaresshadedinsteps1and2togetherformanothersquareofside2cm.

So,itsarea=22cm2. 4. So,from2and3above,1+3=22. 5. Areaofsquaresshadedinsteps1,2,and3=(1+

3+5)cm2. 6. But,thesquaresshadedinsteps1,2and3together

formanothersquareofside3cm. So,itsarea=32cm2. 7. So,from5and6above,1+3+5=32. 8. Areaofthesquaresshadedinsteps1,2,3and4=

(1+3+5+7)cm2. 9. But, the squares shaded in steps 1, 2, 3 and 4

togetherformanothersquareofside4cm. So,itsarea=42cm2. 10. From8and9above,1+3+5+7 = 42. 11. Similarly,wecansaythat1+3+5+7+9+11+

13+15+17+19=102

or1+3+5+…upto10terms=102

or1+3+5…uptonterms=n2

Conclusion :Fromtheaboveactivity,wecansaythatthesumoffirstn oddnaturalnumbersisn2.

Activity-4Objective : Tofind the formula fornth term of anarithmeticprogression.Materials required : Squared paper, colour pencils,geometrybox,etc.

Procedure : LettheAPbe4,6,8,10,12,14,16,18,20. Here,a=4,d=2,andt9 = 20 1. Take a squaredpaper of dimensions 9 cm×20

cm. 2. Shade 4 squares of all the 9 rows as shown in

figure7.

Figure 7 3. Shade1×2=2moresquaresofsecondrowusing

differentcolourasshowninthefigure8.

Figure8 4. Shade2×2=4moresquaresofthirdrowasshown

infigure9.

Figure 9

5. Shade 3 × 2 = 6more squares of fourth row.Continuethisprocessuptothe9throwasshownbelow.

Figure 10

12

GOYAL B

ROTHERS PRAKASHAN

Observations : 1. Infigure10, thefirstrowrepresentsT1 = a = 4 thesecondrowrepresentsT2 = a + d = 4 + 2 thethirdrowrepresentsT3 = a + 2d=4+2×2 thefourthrowrepresentsT4 = a + 3d=4+3×2 thefifthrowrepresentsT5= a + 4d=4+4×2 thesixthrowrepresentsT6= a+5d=4+5×2 theseventhrowrepresentsT7= a+6d=4+6×2 theeighthrowrepresentsT8= a+7d=4+7×2 theninthrowrepresentsT9= a+8d=4+8×2 2. From1above,wehave,T1 = a,T2 = a + d = a +

(2 – 1)d, T3 = a + 2d = a + (3 – 1)d,...T9 = a+8d = a +

(9–1)d Ifwecontinuethepattern,wecanwriteTn= a +

(n – 1)dConclusion : Fromtheaboveactivity,wefindthenthtermofanAPwhosefirsttermisaandcommondifferenced

isgivenbyTn = a + (n – 1)d.Do Yourself :FindtheformulafornthtermofanAPbyactivitymethodtakingthefollowingAP’s

(i) 5,8,11,14,17,… (ii)10,11,12,13,14,…

Activity-5

Objective :Tofindtheformulaforthesumofntermsofanarithmeticprogression.

Materials required : Squaredpaper,colourpencils,geometrybox,apairofscissors,whitesheetsofpaper,gluestick,etc.

Procedure :LetustaketheAP1,4,7,10,13,16,…

Here a=1,d = 3

1. Takeasquaredpaperofsize16cm×6cm.Withthehelpofcolourpencils,shaderectanglesoflength1cm,4cm,7cm,10cm,13cm,16cmandofwidth1cmeachasshowninthefigure.

T1 = 1

T2 = 1 + 3 = 4

T3=1+2×3=7

T4=1+3×3=10

T5=1+4×3=13

T6=1+5×3=16Figure 11

2. Takeanothersquaredpaperofsize16cm×6cm.Shaderectanglesoflength1cm,4cm,7cm,10cm,13cm,16cmandofwidth1cmeachinthereverseorderasshowninfigure12.

Figure 12

3. Now,cutthecolouredportionsoffigures11and12andpastethemonawhitesheetofpaperasshowninfigure.

Figure 13

Observations : 1. Theareaoftheshadedregioninthefigure11 =(1×1+1×4+1×7+1×10+1×13+1×16)cm2

=(1+4+7+10+13+16)cm2

2. Theareaoftheshadedregioninthefigure12 =(1×1+1×4+1×7+1×10+1×13+1×16)cm2

=(1+4+7+10+13+16)cm2

3. Since,figure13ismadeupoftheshadedregionsoffigures11and12,soareaoffigure13

=2(1+4+7+10+13+16)cm2

4. But, figure 13 is a rectangle of dimensions6cm×17cm

So,itsarea=(6×17)cm2

5. From3and4abovewehave,

1+4+7+10+13+16=6

2×17=

6

2(1+16)=

6

2[1+(1+5×3)][ T6=16=1+5×3]

= 6

2[2+5×3]= 6

2[2+(6–1)3]

⇒ T1 + T2 + T3 + T4 + T5 + T6

13

GOYAL B

ROTHERS PRAKASHAN

= 6

2 [2a+(6–1)d]

[Replacing1byaand3byd]

⇒ S6 = T1 + T2 + T3 + T4 + T5 + T6= 6

2

[2a+(6–1)d]

⇒ Sn = T1 + T2 + T3+…Tn = 2

n [2a + (n – 1)d]

[Extendingthepatternfornterms]

6. Fromfigure13,theareaoffirstrow=1×(1+16)cm2

Thereare6rows infigure13.So,areaoffirure13=6×(1+16)cm2

7. From3and6,wehave 2(1+4+7+10+13+16)=6(1+16)

⇒ 1 + 4 +7 + 10 + 13 + 16 = 6

2 (1+16)

⇒ T1 + T2 + T3 + T4 + T5 + T6 = 6

2 (1+16)

= 6

2 (T1 + T6)

⇒ S6 = 6

2 (T1 + T6) ⇒ Sn = 2

n[T1 + Tn]

[Extendingthesamepatternfornterms]

Conclusion : 1. Fromtheaboveactivity,itisverifiedthat (a) thesumoffirstntermsofanAPwhosefirst

termisa andcommondifferenceisd, is given

bySn = 2

n [2a + (n – 1)d]

(b) thesumoffirstntermsofanAPwhosefirsttermisT1andthelasttermisTn,isgivenby

Sn = 2

n [T1 + Tn]