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 _______
Step-by-step solution
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.