Question

1 + 3 + 7 + 15 + 31 + ....... to n terms =

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

Let $T_n$ be the $n^{th}$ term and S the sum upto n terms.

S = 1 + 3 + 7 + 15 + 31 + ..........+ $T_n$

Again S = 1 + 3 + 7 + 15 + .......... + $T_{n-1}+T_n$

Subtracting, we get 0 = 1 + {2 + 4 + 8 + .........($T_n-T_{n-1}$)} - $T_n$

∴ $T_n$ = 1 + 2 + $2^2$ + $2^3$ + ......... upto n terms

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

Now S = $\sum T_n$ = $\sum 2^n$ - $\sum$1

= (2 + $2^2$ + $2^3$ + ...........+ $2^n$) - n

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

Aliter : 1 + 3 + 7 + ........ + $T_n$

= 2 - 1 + $2^2$ - 1 + $2^3$ - 1 + ........ + $2^n$ - 1

= (2 + $2^2$ + ......... + $2^n$) - n = $2^{n+1}$ - 2 - n

Trick: Check the options for n = 1, 2.