Arithmetic progression

ARITHMETIC PROGRESSION

EX. NO. 2

1. Which of the following lists of numbers are Arithmetic Progressions? Justify.

(i) 1, 3, 6, 10, .....Let, t1 = 1, t2 = 3, t3 = 6, t4 = 10t2 – t1 = 3 – 1 = 2t3 – t2 = 6 – 3 = 3t4 – t3 = 10 – 6 = 4As, The difference between two consecutive terms is not constant.So, The sequence is not an A.P.

ii. 3, 5, 7, 9, 11, .....

Sol. Let, t1 = 3, t2 = 5, t3 = 7, t4 = 9, t5 = 11t2 – t1 = 5 – 3 = 2t3 – t2 = 7 – 5 = 2t4 – t3 = 9 – 7 = 2t5 – t4 = 11 – 9 = 2As, The difference between two consecutive terms is 2 which is constant.So, The sequence is an A.P.

(iii) 1, 4, 7, 10, ....

Sol. Let, t1 = 1, t2 = 4, t3 = 7, t4 = 10t2 – t1 = 4 – 1 = 3t3 – t2 = 7 – 4 = 3t4 – t3 = 10 – 7 = 3As, The difference between two consecutive terms 3 which is constant.So, The sequence is an A.P.

(iv) 3, 6, 12, 24, .... Sol. t1 = 3, t2 = 6, t3 = 12, t4 = 24t2 – t1 = 6 – 3 = 3t3 – t2 = 12 – 6 = 6t4 – t3 = 24 – 12 = 12 Since, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.

(v) 22, 26, 28, 31, ... Sol. t1 = 22, t2 = 26, t3 = 28, t4 = 31t2 – t1 = 26 – 22 = 4t3 – t2 = 28 – 26 = 2t4 – t3 = 21 – 28 = 3Since, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.

(vi) 0.5, 2, 3.5, 5, ... Sol. t1 = 0.5, t2 = 2, t3 = 3.5, t4 = 5t2 – t1 = 2 – 0.5 = 1.5t3 – t2 = 3.5 – 2 = 1.5t4 – t3 = 5 – 3.5 = 1.5As, The difference between two consecutive terms is 1.5 which is constant.So, The sequence is an A.P.

(vii) 4, 3, 2, 1, .... Sol. t1 = 4, t2 = 3, t3 = 2, t4 = 1,t2 – t1 = 3 – 4 = – 1t3 – t2 = 2 – 3 = – 1t4 – t3 = 1 – 2 = – 1As, The difference between two consecutive terms is –1 which is constant.So, The sequence is an A.P.

(viii) – 10, – 13, – 16, – 19, .....

Sol. t1 = – 10, t2 = – 13, t3 = – 16, t4 = – 19t2 – t1 = – 13 – (– 10) = – 13 + 10 = – 3t3 – t2 = – 16 – (– 13) = – 16 + 13 = – 3t4 – t3 = – 19 – (– 16) = – 19 + 16 = – 3As, The difference between two consecutive terms is – 3 which is constant. So, The sequence is an A.P.

2. Write the first five terms of the following Arithmetic Progressions where,the common difference ‘d’ and the first term ‘a’ are given :(i) a = 2, d = 2.5Sol. a = 2, d = 2.5Here, t1 = a = 2t2 = t1 + d = 2 + 2.5 = 4.5t3 = t2 + d = 4.5 + 2.5 = 7t4 = t3 + d = 7 + 2.5 = 9.5t5 = t4 + d = 9.5 + 2.5 = 12Thus, The first five terms of the A.P. are 2, 4.5, 7, 9.5 and 12.

(ii) a = 10, d = – 3 Sol. a = 10, d = – 3Here, t1 = a = 10t2 = t1 + d = 10 + (– 3) = 10 – 3 = 7t3 = t2 + d = 7 + (– 3) = 7 – 3 = 4t4 = t3 + d = 4 + (– 3) = 4 – 3 = 1t5 = t4 + d = 1 + (– 3) = 1 – 3 = – 2Thus, The first five terms of the A.P. are 10, 7, 4, 1 and – 2.

(iii) a = 4, d = 0Sol. a = 4, d = 0Here, t1 = a = 4t2 = t1 + d = 4 + 0 = 4t3 = t2 + d = 4 + 0 = 4t4 = t3 + d = 4 + 0 = 4t5 = t4 + d = 4 + 0 = 4Thus, The first five terms of the A.P. are 4, 4, 4, 4 and 4.

(iv) a = 5, d = 2Sol. a = 5, d = 2Here, t1 = a = 5t2 = t1 + d = 5 + 2 = 7t3 = t2 + d = 7 + 2 = 9t4 = t3 + d = 9 + 2 = 11t5 = t4 + d = 11 + 2 = 13Thus, The first five terms of the A.P. are 5, 7, 9, 11 and 13.

(v) a = 3, d = 4 Sol. a = 3, d = 4Here, t1 = a = 3t2 = t1 + d = 3 + 4 = 7t3 = t2 + d = 7 + 4 = 11t4 = t3 + d = 11 + 4 = 15t5 = t4 + d = 15 + 4 = 19Thus, The first five terms of the A.P. are 3, 7, 11, 15 and 19.

(vi) a = 6, d = 6

Sol. a = 6, d = 6Here, t1 = a = 6t2 = t1 + d = 6 + 6 = 12t3 = t2 + d = 12 + 6 = 18t4 = t3 + d = 18 + 6 = 24t5 = t4 + d = 24 + 6 = 30Thus,The first five terms of A.P. are 6, 12, 18, 24 and 30.