Show that in an infinite G.P. with common ratio r (|r|<1 ), each … - Vidyadip

Question

Show that in an infinite G.P. with common ratio $\text{r}\big(|\text{r}|<1\big),$ each terms bears a constant ratio to the sum of all terms that follow it.

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Answer

Let a be first term and r be common ratio of G.P. Here, $\frac{\text{a}_\text{n}}{\big(\text{a}_{\text{n}+1}+\text{a}_{\text{n}+2}+\ \dots\infty\big)}=\frac{\text{ar}^{\text{n}-1}}{\text{ar}^{\text{n}}+\text{ar}^{\text{n}+1}+\ \dots}$ $=\frac{\text{ar}^{\text{n}-1}}{\text{ar}^{\text{n}}\big(1+\text{r}+\text{r}^2+\ \dots\infty\big)}$ $=\frac{\text{ar}^{\text{n}-1}}{\text{ar}^{\text{n}}\Big(\frac{1}{1-\text{r}}\Big)}$ $=\Big(\frac{1-\text{r}}{\text{r}}\Big)$ Since r is a constant, so $\Big(\frac{\text{a}_\text{n}}{\text{a}_{\text{n}+1}+\text{a}_{\text{n}+2}+\ \dots\infty}\Big)=\text{k}\ (\text{constant})$ Such that $\text{k}=\Big(\frac{1-\text{r}}{\text{r}}\Big)$

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