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

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 &#8804