AS Maths Masterclass

Lesson 1:

Arithmetic series

Learning objectives

The student should be able to:

• recognise an Arithmetic Progression (AP);

• recall the formula for the sum to n terms;

• evaluate the terms and sum of a given AP;

• manipulate formulae that model APs.

What do the following have in common ?

5 + 7 + 9 + 11 + 13 + … … + 29

– 8 – 5 – 2 + 1 + 4

40 + 30 + 20 + 10 + 0 – 10 – 20 – 30 – 40

“They all have a difference (d) in common!”

E.g. Take 5 + 7 + 9 + 11 + … + 29

Each term is bigger than its previous term by 2

So

Also

In general

Or

212 uu

223 uu

21 nn uu

duu nn 1

“Let’s go straight to the nth term”

We have that

And that

And further that

In general:

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daddaduu 2)(23

daddaduu 3)2(34

daddaduu 4)3(45

1)d(naun

Proof of the sum to n terms

If we write out the terms of the series we get

If we now write out these terms in reverse order

Adding each pair of terms we then get

And so

nS a )( da )2( da )]1[(... dna

nS )]1[( dna )]2[( dna a ...)]3[( dna

nS2 )]1[2( dna nnS 2

n1]d)[n(2a

Finding a formula for

First take the sum formula:

Then substitute a = 1, d = 1 to get

So 1+2+3+…+100 = 50 x 101 = 5050 etc

1]d)[n(2a 2

nnS

]1)1(12[21

nn

rn

r

]12[2

nn 1)(n

2

n

n

rr1

Arithmetic Progression Example

The 5th term of an AP is and the 7th term of the same AP is Find a and d.

Well, writing down the nth terms (n = 5,7) gives

Subtracting gives from which

Substituting this in either equation leads to

61

21

21

6,61

4 dada

31

2 d61

d

21

a