Sum the series: 1+3+7+15+31+ to n terms - Maths - Sequences and Series

Question

Sum the series: 1+3+7+15+31+ to n terms

Santosh Kumar · Accepted Answer

Let sn = 1 + 3 + 7 + 15 + 31 + 63 nth term, tn = 2n â€“ 1 $s_{n} = Σ_{i = 1}^{n} (t_{n}) = Σ_{i = 1}^{n} (2^{n} - 1) = Σ_{i = 1}^{n} 2^{n} - Σ_{i = 1}^{n} 1$ = (21 + 22 + 23 + + 2n) â€“ (1 + 2 + n) $= \frac{2^{1} (2^{n} - 1)}{2 - 1} - \frac{n (n + 1)}{2} = 2 (2^{n} - 1) - \frac{n (n + 1)}{2}$

Sum the series: 1+3+7+15+31+............to n terms.