Question
If $2^{n+2}-2^{n+1}+2^n=c \times 2^n$, find the value of $c$.
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Answer
$\text { We have, } 2^{n+2}+2^{n+1}+2 n=c \times 2^n$
$\Rightarrow 2^n 2^2+2^n 2^1+2^n$
$=c \times 2^n\left[\therefore a^{m+n}=a^m \times a^n\right]$
$\Rightarrow 2^n\left[2^2-2^1+1\right]$
$=c \times 2^n\left[\text { taking common } 2^n \text { in LHS }\right]$
$\Rightarrow 2^n[4-2+1]=c \times 2^n$
$\Rightarrow 3 \times 2^n=c \times 2^n 3 \times 2^n \times 2^{-n}$
$=c \times 3^n \times c^{-n}\left[\text { multiplying both sides by } 2^{-n}\right]$
$\Rightarrow 3 \times 2^{n-1}=c \times 2^{n-n}\left[\therefore a^{m+n}=a^m \times a^n\right]$
$\Rightarrow 3 \times 2^0=c \times 2^0$
$\Rightarrow 3 \times 1=c \times 1\left[\therefore a^0=1\right]$
$\therefore 3=c$
$\Rightarrow 2^n 2^2+2^n 2^1+2^n$
$=c \times 2^n\left[\therefore a^{m+n}=a^m \times a^n\right]$
$\Rightarrow 2^n\left[2^2-2^1+1\right]$
$=c \times 2^n\left[\text { taking common } 2^n \text { in LHS }\right]$
$\Rightarrow 2^n[4-2+1]=c \times 2^n$
$\Rightarrow 3 \times 2^n=c \times 2^n 3 \times 2^n \times 2^{-n}$
$=c \times 3^n \times c^{-n}\left[\text { multiplying both sides by } 2^{-n}\right]$
$\Rightarrow 3 \times 2^{n-1}=c \times 2^{n-n}\left[\therefore a^{m+n}=a^m \times a^n\right]$
$\Rightarrow 3 \times 2^0=c \times 2^0$
$\Rightarrow 3 \times 1=c \times 1\left[\therefore a^0=1\right]$
$\therefore 3=c$
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