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Q.
If $f(1)=1$ and $f(n+1)=2f (n)+1$ if $n \ge1$, then $f(n)$ is equal to
Relations and Functions
Solution:
$f\left(1\right)=1$ 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, $f\left(1\right)=2^{1}-1, f\left(2\right)=2^{2}-1$,
$f\left(3\right) = 2^{3} - 1, \ldots, f\left(n\right)$
$=2^{n}-1$.