Question Number 145499 by mathdanisur last updated on 05/Jul/21
$$\frac{\mathrm{1}}{\mathrm{2}}\:-\:\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{1}}{\mathrm{8}}\:-\:\frac{\mathrm{1}}{\mathrm{16}}\:+\:…\:-\:\frac{\mathrm{1}}{\mathrm{256}}\:=\:\frac{\boldsymbol{{z}}+\mathrm{1}}{\mathrm{256}} \\ $$$${find}\:\:\boldsymbol{{z}}=? \\ $$
Answered by gsk2684 last updated on 05/Jul/21
$$\mathrm{sum}\:\mathrm{of}\:\mathrm{8}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{G}.\mathrm{P}.\: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}−\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{8}} }{\mathrm{1}−\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)}\right)=\frac{\mathrm{84}+\mathrm{1}}{\mathrm{256}}\: \\ $$
Commented by mathdanisur last updated on 05/Jul/21
$${thanks}\:{Ser}\:{cool} \\ $$
Answered by Ar Brandon last updated on 05/Jul/21
$$\mathrm{S}=−\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{8}} {\sum}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{n}} =−\frac{−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{8}} \right)}{\mathrm{1}−\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$$\:\:\:=\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{256}}\right)}{\frac{\mathrm{3}}{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{255}}{\mathrm{256}}\right)=\frac{\mathrm{z}+\mathrm{1}}{\mathrm{256}} \\ $$$$\Rightarrow\mathrm{z}=\frac{\mathrm{255}}{\mathrm{3}}−\mathrm{1}=\frac{\mathrm{252}}{\mathrm{3}}=\mathrm{84} \\ $$
Commented by mathdanisur last updated on 05/Jul/21
$${thanks}\:{Ser}\:{cool} \\ $$