# JEE Main & Advanced Mathematics Geometric Progression ${{n}^{th}}$term of A.G.P.

## ${{n}^{th}}$term of A.G.P.

Category : JEE Main & Advanced

If ${{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,......,\,{{a}_{n}},\,......$ is an A.P. and ${{b}_{1}},\,{{b}_{2}},\,\,......,\,{{b}_{n}},\,......$ is a G.P., then the sequence ${{a}_{1}}{{b}_{1}},\,{{a}_{2}}{{b}_{2}},\,{{a}_{3}}{{b}_{3}},$$\,......,\,{{a}_{n}}{{b}_{n}},\,.....$ is said to be an arithmetico-geometric sequence.

Thus, the general form of an arithmetico geometric sequence is $a,\,(a+d)\,r,\,(a+2d)\,{{r}^{2}},\,(a+3d)\,{{r}^{3}},\,.....$

From the symmetry we obtain that the nth term of this sequence is $[a+(n-1)\,d]\,{{r}^{n-1}}$.

Also, let $a,\,(a+d)\,r,\,(a+2d)\,{{r}^{2}},\,(a+3d)\,{{r}^{3}},\,.....$be an arithmetico-geometric sequence.

Then, $a+\,(a+d)\,r$$+\,(a+2d)\,{{r}^{2}}+(a+3d)\,{{r}^{3}}+...$ is an arithmetico-geometric series.

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