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Question Number 1789 by 112358 last updated on 26/Sep/15

If ∣r∣≠1, find an expression for T_n (r), where T_n (r)=1+r^2 +r^3 +r^4 +r^6 +r^8 +r^9 +r^(10) +r^(12) +r^(14) +r^(15) +r^(16) + ...+r^(6n) .

$${If}\:\mid{r}\mid\neq\mathrm{1},\:{find}\:{an}\:{expression}\:{for} \\ $$$${T}_{{n}} \left({r}\right),\:{where}\: \\ $$$${T}_{{n}} \left({r}\right)=\mathrm{1}+{r}^{\mathrm{2}} +{r}^{\mathrm{3}} +{r}^{\mathrm{4}} +{r}^{\mathrm{6}} +{r}^{\mathrm{8}} +{r}^{\mathrm{9}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+{r}^{\mathrm{10}} +{r}^{\mathrm{12}} +{r}^{\mathrm{14}} +{r}^{\mathrm{15}} +{r}^{\mathrm{16}} + \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...+{r}^{\mathrm{6}{n}} \:. \\ $$

Answered by Rasheed Soomro last updated on 27/Sep/15

T_n (r)= {1+r^2 +r^4 +...+r^(6n) }+{1+r^3 +r^6 +...+r^(6n) }−{1+r^6 +r^(12) +...+r^(6n) } Formula: S=a+ar+ar^2 +...+lr^(−1) +l=((rl−a)/(r−1)) ={((r^2 .r^(6n) −1)/(r^2 −1))}+{((r^3 .r^(6n) −1)/(r^3 −1))}−{((r^6 .r^(6n) −1)/(r^6 −1))} =((r^(6n+2) −1)/(r^2 −1))+((r^(6n+3) −1)/(r^3 −1))−((r^(6n+6) −1)/(r^6 −1))

$${T}_{{n}} \left({r}\right)= \\ $$$$\left\{\mathrm{1}+{r}^{\mathrm{2}} +{r}^{\mathrm{4}} +...+{r}^{\mathrm{6}{n}} \right\}+\left\{\mathrm{1}+{r}^{\mathrm{3}} +{r}^{\mathrm{6}} +...+{r}^{\mathrm{6}{n}} \right\}−\left\{\mathrm{1}+{r}^{\mathrm{6}} +{r}^{\mathrm{12}} +...+{r}^{\mathrm{6}{n}} \right\} \\ $$$${Formula}: \\ $$$${S}={a}+{ar}+{ar}^{\mathrm{2}} +...+{lr}^{−\mathrm{1}} +{l}=\frac{{rl}−{a}}{{r}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:=\left\{\frac{{r}^{\mathrm{2}} .{r}^{\mathrm{6}{n}} −\mathrm{1}}{{r}^{\mathrm{2}} −\mathrm{1}}\right\}+\left\{\frac{{r}^{\mathrm{3}} .{r}^{\mathrm{6}{n}} −\mathrm{1}}{{r}^{\mathrm{3}} −\mathrm{1}}\right\}−\left\{\frac{{r}^{\mathrm{6}} .{r}^{\mathrm{6}{n}} −\mathrm{1}}{{r}^{\mathrm{6}} −\mathrm{1}}\right\} \\ $$$$\:\:\:\:\:=\frac{{r}^{\mathrm{6}{n}+\mathrm{2}} −\mathrm{1}}{{r}^{\mathrm{2}} −\mathrm{1}}+\frac{{r}^{\mathrm{6}{n}+\mathrm{3}} −\mathrm{1}}{{r}^{\mathrm{3}} −\mathrm{1}}−\frac{{r}^{\mathrm{6}{n}+\mathrm{6}} −\mathrm{1}}{{r}^{\mathrm{6}} −\mathrm{1}} \\ $$$$ \\ $$

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