Question

If the sum of 1 + 2 + 2² + ....... + 2^{n – 1} is 255, find the value of n.

Answer 1

Required series: 1 + 2 + 2² + ....... + 2^{n – 1}

Now, `2/1 = 2, 2^2/2 = 2`

Thus, given series is a G.P. with first term, a = 1

Common ratio, r = 2  ...(∵ r > 1)

Last term, l = 2^{n – 1}

Let there be n terms in the series.

Then, S_n = 255

`=> (lr - a)/(r - 1) = 255`

`=> (2^(n - 1) xx 2 - 1)/(2 - 1) = 255`

`=>` 2^{n – 1} × 2 – 1 = 255

`=>` 2^{n – 1} × 2 = 256

`=>` 2^{n – 1} = 128

`=>` 2^{n – 1} = 2⁷

`=>` n – 1 = 7

`=>` n = 8