Question Number 1789 by 112358 last updated on 26/Sep/15
$${If}\:\mid{r}\mid\neq\mathrm{1},\:{find}\:{an}\:{expression}\:{for} \\ $$$${T}_{{n}} \left({r}\right),\:{where}\: \\ $$$${T}_{{n}} \left({r}\right)=\mathrm{1}+{r}^{\mathrm{2}} +{r}^{\mathrm{3}} +{r}^{\mathrm{4}} +{r}^{\mathrm{6}} +{r}^{\mathrm{8}} +{r}^{\mathrm{9}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+{r}^{\mathrm{10}} +{r}^{\mathrm{12}} +{r}^{\mathrm{14}} +{r}^{\mathrm{15}} +{r}^{\mathrm{16}} + \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…+{r}^{\mathrm{6}{n}} \:. \\ $$
Answered by Rasheed Soomro last updated on 27/Sep/15
$${T}_{{n}} \left({r}\right)= \\ $$$$\left\{\mathrm{1}+{r}^{\mathrm{2}} +{r}^{\mathrm{4}} +…+{r}^{\mathrm{6}{n}} \right\}+\left\{\mathrm{1}+{r}^{\mathrm{3}} +{r}^{\mathrm{6}} +…+{r}^{\mathrm{6}{n}} \right\}−\left\{\mathrm{1}+{r}^{\mathrm{6}} +{r}^{\mathrm{12}} +…+{r}^{\mathrm{6}{n}} \right\} \\ $$$${Formula}: \\ $$$${S}={a}+{ar}+{ar}^{\mathrm{2}} +…+{lr}^{−\mathrm{1}} +{l}=\frac{{rl}−{a}}{{r}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:=\left\{\frac{{r}^{\mathrm{2}} .{r}^{\mathrm{6}{n}} −\mathrm{1}}{{r}^{\mathrm{2}} −\mathrm{1}}\right\}+\left\{\frac{{r}^{\mathrm{3}} .{r}^{\mathrm{6}{n}} −\mathrm{1}}{{r}^{\mathrm{3}} −\mathrm{1}}\right\}−\left\{\frac{{r}^{\mathrm{6}} .{r}^{\mathrm{6}{n}} −\mathrm{1}}{{r}^{\mathrm{6}} −\mathrm{1}}\right\} \\ $$$$\:\:\:\:\:=\frac{{r}^{\mathrm{6}{n}+\mathrm{2}} −\mathrm{1}}{{r}^{\mathrm{2}} −\mathrm{1}}+\frac{{r}^{\mathrm{6}{n}+\mathrm{3}} −\mathrm{1}}{{r}^{\mathrm{3}} −\mathrm{1}}−\frac{{r}^{\mathrm{6}{n}+\mathrm{6}} −\mathrm{1}}{{r}^{\mathrm{6}} −\mathrm{1}} \\ $$$$ \\ $$