Question Number 92324 by I want to learn more last updated on 06/May/20
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{x}\:\:\mathrm{for}\:\mathrm{which}\:\:\:\:\underset{\mathrm{n}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}\:\:=\:\:\infty} {\sum}}\:\mathrm{16}\left(\frac{\mathrm{3}}{\mathrm{4}}\mathrm{x}\:\:+\:\:\mathrm{1}\right)^{\mathrm{n}} \\ $$$$\left(\mathrm{a}\right)\:\:\:\mathrm{Is}\:\mathrm{convergent} \\ $$$$\left(\mathrm{b}\right)\:\:\:\mathrm{Is}\:\mathrm{equal}\:\mathrm{to}\:\:\mathrm{10}\frac{\mathrm{2}}{\mathrm{3}} \\ $$
Answered by mr W last updated on 06/May/20
$${convergent}\:{if} \\ $$$$\mid\frac{\mathrm{3}}{\mathrm{4}}{x}+\mathrm{1}\mid<\mathrm{1} \\ $$$$−\mathrm{1}<\frac{\mathrm{3}}{\mathrm{4}}{x}+\mathrm{1}<\mathrm{1} \\ $$$$\Rightarrow−\frac{\mathrm{8}}{\mathrm{3}}<{x}<\mathrm{0} \\ $$$$ \\ $$$${S}=\mathrm{16}×\frac{\mathrm{1}}{\mathrm{1}−\left(\frac{\mathrm{3}}{\mathrm{4}}{x}+\mathrm{1}\right)}=\frac{\mathrm{32}}{\mathrm{3}} \\ $$$$\Rightarrow{x}=−\mathrm{2} \\ $$
Commented by I want to learn more last updated on 06/May/20
$$\mathrm{Thanks}\:\mathrm{sir} \\ $$