Question

If $f(1)=1$ and $f(n+1)=2f (n)+1$ if $n \ge1$, then $f(n)$ is equal to

• A. $2^{n}+1$
• B. $2^{n}$
• C. $2^{n}-1$
• D. $2^{n-1}-1$

Answer 1

Answer: (C)

$2^{n}-1$

Answer 2

$f\left(1\right)=1$ and $f\left(n+1\right)=2f\left(n\right)+1, n \ge1$. $\therefore f\left(2\right)=2\left(1\right)+1=3, f\left(3\right)=7, f\left(4\right)=15, \ldots$ and so on Thus, $f\left(1\right)=2^{1}-1, f\left(2\right)=2^{2}-1$, $f\left(3\right) = 2^{3} - 1, \ldots, f\left(n\right)$ $=2^{n}-1$.