JEE Main & Advanced Mathematics Sequence & Series Sum of A.G.P.

(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 \[{{S}_{\infty }}=\frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}}\].