Question

The sum to n terms of the series $1 + 3 + 7 + 15 + 31 + \ldots \ldots$ is

• A. $2 ^ { n + 1 } - n$
• B. $2 ^ { n + 1 } - n - 2$
• C. $2 ^ { n } - n - 2$
• D. None of these

Answer 1

Answer: (B)

$2 ^ { n + 1 } - n - 2$

Answer 2

\begin{tabular} { l } $S = 1 + 3 + 7 + 15 + 31 + \ldots .$. \ $S = 1 + 3 + 7 + 15 + \ldots \ldots$. \ \hline $0 = \left( 1 + 2 + 4 + 8 + 16 + \ldots . \right.$. to $n$ terms) $- T _ { n } \quad$ (on subtracting) \end{tabular}

∴ $T _ { n } = 1 + 2 + 4 + 8 + \ldots \ldots \ldots \ldots$. . to $n$ terms $= 1 \cdot \frac { 2 ^ { n } - 1 } { 2 - 1 } = 2 ^ { n } - 1$ $S _ { n } = \sum _ { n = 1 } ^ { n } T _ { n } = \sum _ { n = 1 } ^ { n } \left( 2 ^ { n } - 1 \right) = \sum _ { n = 1 } ^ { n } 2 ^ { n } - \sum _ { n = 1 } ^ { n } 1 = 2 \cdot \left( \frac { 2 ^ { n } - 1 } { 2 - 1 } \right) - n = 2 ^ { n + 1 } - n - 2$