Question

Let a₁, a₂, a₃, … be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b₁, b₂, b₃, … be a sequence of positive integers in geometric progression with common ratio 2. If a₁ = b₁ = c, then the number of all possible values of c, for which the equality 

2(a₁ + a₂ + ... + a_n) = b₁ + b₂ + … + b_n 

holds for some positive integer n, is ___.

Answer 1

Given 2(a₁ + a₂ + ... + a_n) = b₁ + b₂ + … + b_n 

Checking c against these values of n 

We get c = 12 (when n = 3) 

Hence number of such c = 1

Answer 2

Given,

a₁, a₂, a₃, ....is A.P with common difference d = 2.

b₁, b₂, b₃, .....is G.P with common ratio r = 2

a₁ = b₁ = c holds for some positive integer n where

n ≤ 6

Since n is positive integer, for i.e n ={1,2,3,4,5,6} inequality holds

n = 1 ⇒ c = 0 Rejected c ≥ 1

n = 2 ⇒ c < 0 Rejected c ≥ 1

n = 3 ⇒ c = \(\frac{6-18}{6-8+1} = 12\) correct

n = 4 ⇒ c = not an integer Rejected

n = 5 ⇒ c = not an integer Rejected

n = 6 ⇒ c = not an integer Rejected

Therefore, c = 12 for n = 3

Hence, the number of c holds for some positive integer n where 2(a₁ + a₂ + a₃ + ..... + a_n) = b₁ + b₂ + b₃ + , ..... + b_n is one.