Question

Question 90:

If $2^{n+2}-2^{n+1}+2^n=C\times 2^n,thenfindC.$

Answer 1

We have,
$2^{n+2}-2^{n+1}+2^n=C\times 2^n$
$2^{n+2}-2^{n+1}+2^n=C\times 2^n\Rightarrow 2^n{.2}^2-2^n{.2}^1+2^n=C\times 2^n[\because a^{m+n}=a^m\times a^n]\Rightarrow 2^n[2^2-2^1+1]=C\times 2^n\left[takingcommon{2}^{n}inLHS\right]\Rightarrow 2^n[4-2+1]=C\times 2^n\Rightarrow 3\times 2^n=C\times 2^{n}3\times 2^n\times 2^{-n}=C\times 2^n\times 2^{-n}\left[multiplyingbothsidesby{2}^{-n}\right]\Rightarrow 3\times 2^{n-n}=C\times 2^{n-n}[\because a^m\times a^n=a^{m+n}]\Rightarrow 3\times 2^0=C\times 2^0\Rightarrow 3\times 1=C\times 1\therefore 3=C[\because a^0=1]$