# Evaluate the product prod_(k=1)^n 2*4^k

Evaluate the product $\prod _{k=1}^{n}2\ast {4}^{k}$
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iljovskint
$\prod _{k=1}^{n}2\ast {4}^{k}={2.4}^{1}\ast {2.4}^{2}...2{.4}^{n}={2}^{n}\ast {4}^{1+2+3+...+n}={2}^{n}{4}^{\frac{n\left(n+1\right)}{2}}=$
${2}^{n}\left({2}^{2}{\right)}^{\frac{n\left(n+1\right)}{2}}={2}^{n}{2}^{n\left(n+1\right)}={2}^{{n}^{2}+2n}$

Nash Frank
$\prod _{k=1}^{n}﻿2\ast {4}^{k}=\left[2\ast 2...\ast 2\right]\left[4\ast {4}^{2}\ast {4}^{3}\ast ....\ast {4}^{n}\right]={2}^{n}\left[{4}^{1+2+...+n}\right]={2}^{n}\left[{4}^{\sum _{i=1}^{n}i}\right]$
$={2}^{n}\left[{4}^{\frac{n\left(n+1\right)}{2}}\right]={4}^{\frac{1}{2}n}\ast {4}^{\frac{n\left(n+1\right)}{2}}={4}^{\frac{1}{2}\left(n\left(n+1\right)+n\right)}={2}^{{n}^{2}+2n}={2}^{n\left(n+2\right)}$