Let a 1 , a 2 , a 3 , . . . . . . . . . be a sequence of positive integers in arithmetic progression with common difference 2 . Also, let b 1 , b 2 , b 3 , . . . . . . . . be a sequence of positive in

Question

Let a 1 , a 2 , a 3 , . . . . . . . . . be a sequence of positive integers in arithmetic progression with common difference 2 . Also, let b 1 , b 2 , b 3 , . . . . . . . . be a sequence of positive integers in geometric progression with common ratio 2 . If a 1 = b 1 = c , then the number of all possible values of c , for which the equality 2 a 1 + a 2 + . . . . . . . . + a n = b 1 + b 2 + . . . . . . . . + b n holds for some positive integer n , is _______

TestHub · Accepted Answer

2 a 1 + a 2 + . . . . . . . . + a n = b 1 + b 2 + . . . . . . . . + b n ⇒ 2 n 2 2 a 1 + n − 1 ⋅ 2 = b 1 ⋅ 2 n − 1 2 − 1 ⇒ n 2 c + 2 n − 2 = c 2 n − 1 ⇒ 2 n c + n − 1 = c 2 n − 1 ⇒ c 2 n − 2 n − 1 = 2 n 2 − 2 n ⇒ c = 2 n 2 − 2 n 2 n − 2 n − 1 ≥ 1 . . . . . . . . . . . . . i ⇒ 2 n 2 − 2 n ≥ 2 n − 2 n − 1 ⇒ 2 n 2 + 1 ≥ 2 n ⇒ n ≤ 6 Now put n = 1 , 2 , . . . . . . , 6 in equation i and using c ∈ I + , we get c = 12 , when n = 3 So, only one value of c is possible.

Mathematics - Sequence & Series Question with Solution | TestHub

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