# JEE Main & Advanced Mathematics Geometric Progression Sum of A.G.P.

## Sum of A.G.P.

Category : JEE Main & Advanced

(1) Sum of $n$ terms : The sum of n terms of an arithmetico-geometric sequence $a,\,(a+d)\,r,\,(a+2d)\,{{r}^{2}},\,$$(a+3d)\,{{r}^{3}},\,.....$ is given by  {{S}_{n}}=\left\{ \begin{align}& \frac{a}{1-r}+dr\frac{(1-{{r}^{n-1}})}{{{(1-r)}^{2}}}-\frac{\{a+(n-1)\,d\}{{r}^{n}}}{1-r},\text{ when }r\ne 1 \\& \frac{\text{n}}{\text{2}}[2a+(n-1)\,d],\text{ when }r=1 \\\end{align} \right.\text{ }

(2) Sum of infinite sequence: Let $|r|\,<1$. Then ${{r}^{n}},\,{{r}^{n-1}}\to 0$ as $n\to \infty$ and it can also be shown that $n\,.\,{{r}^{n}}\to 0$ as $n\to \infty$. So, we obtain that ${{S}_{n}}\to \frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}}$, as $n\to \infty$¥.

In other words, when  $|r|\,<1$ the sum to infinity of an arithmetico-geometric series is ${{S}_{\infty }}=\frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}}$.

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