Question

The common ratio, last term and sum of n terms of a G.P. are 2, 128 and 255, respectively. Find the value of n.

Answer 1

r = 2 > 1

a_n = 128

S_n = 255

`a_n = ar^(n - 1)`

128 = `a(2)^(n - 1)`

a = `128/2^(n - 1)`

`S_n = (a(r^n - 1))/(r - 1)`

255 = `(a(2^n - 1))/(2 - 1)`

255 = `128/2^(n - 1)(2^n - 1)`

255 = `128[2^(n - n + 1) - 1/(2^(n - 1))]`

`255/128 = 2 - 1/2^(n - 1)`

`1/2^(n - 1) = 2 - 255/128`

`1/2^(n - 1) = (256 - 255)/128`

`1/2^(n - 1) = 1/128`

`1/2^(n - 1) = 1/2^7`

`2^(n - 1) = 2^7`

n − 1 = 7

n = 8