Question

If $f\left(1\right)=1$ and $f(x+1)=2f\left(x\right)+1$ for $x\ge 1$, then find the range of $f\left(x\right)$.

Answer 1

$f(1+1)=2f\left(1\right)+1=3(\because f(1)=1)$
$f(2+1)=2f\left(2\right)+1=2\times \left(3\right)+1=7$,
$f(3+1)=2f\left(3\right)+1=2\times \left(7\right)+1=15$
The required range is 1, 3, 7, 15, 31, ...........