Question Number 165210 by mkam last updated on 27/Jan/22
$$\underset{\boldsymbol{{n}}=\mathrm{4}} {\overset{\infty} {\sum}}\:\mathrm{2}\:×\:\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\boldsymbol{{n}}+\mathrm{3}} \\ $$
Commented by cortano1 last updated on 27/Jan/22
![2×[((3/4))^7 +((3/4))^8 +... ] = 2×((((3/4))^7 )/(((1/4)))) = 8×((3/4))^7](Q165211.png)
$$\mathrm{2}×\left[\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{7}} +\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{8}} +...\:\right]\:=\:\mathrm{2}×\frac{\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{7}} }{\left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$$$\:=\:\mathrm{8}×\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{7}} \\ $$