JEE Main & Advanced Mathematics Sequence & Series \[{{n}^{th}}\]term of A.G.P.

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.