Question

If $F(n+1)=\frac{2F\left(n\right)+1}{2}$ for $n=1,2,\dots$ and $F\left(1\right)=2$, then $F\left(101\right)$ equals

Answer 1

$F(n+1)=\frac{2F\left(n\right)+1}{2}$
$\Rightarrow F(n+1)-F\left(n\right)=\frac{1}{2}$
Putting $n=1,2,\dots ,100$ and adding, we get
$F\left(101\right)-F\left(1\right)=100\times \frac{1}{2}=50$
$F\left(101\right)=52(\because F(1)=2)$