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 $f\left(x\right)$

• A. $R$
• B. A series like $1,3,5,7,9,...$
• C. A series like $1,3,7,15,...$
• D. $R-\{1,3,7,15,...\}$

Answer 1

Answer: (C)

A series like $1,3,7,15,...$

Given, $f\left(1\right)=1$ and $f(x+1)=2f\left(x\right)+1$

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

Hence, option C is correct.