Question

The sum to n terms of the series 1 + 3 + 7 + 15 + . . . . . .

Answer 1

First step in solving this problem is finding a pattern in the given numbers. They are not in G.P. or A.P.

If you take the difference of the terms you get the numbers 1, (3 - 1), (7 - 3) and (15 - 7) or 1, 2, 4 and 8. These terms are

In G.P. You will learn how to guess/ proceed after this in the section method of difference. in this problem, it is

enough to know that they are related to G.P. or derived from G.P. Now you have to find how we can arrive at the

given series from the G.P. 1, 2, 4, 8, 16........ We get it by subtracting 1 from the given series. First term 0(1 - 1) is not

there. But it does not make any difference.

$\Rightarrow$The series can be written as

S = ($2^1$ - 1) + ($2^2$ - 1) + ($2^3$ - 1) + ($2^4$ - 1) ...............n terms

S = $2^1$ + $2^2$ .............$2^n$ - (1 + 1 + ...........n terms)

= $\frac{2(n^1-1)}{2-1}$ - n

= $2^{n-1}$ - n - 2 - D gives right answer

You can solve this by sustituting n = 1, 2, 3........ and eliminating the options.