Question
In a, G.P. of $$3n$$ terms, $$S_1$$ denotes the sum of first $$n$$ terms, $$S_2$$ the sum of the second block of $$n$$ terms and $$S_3$$ the sum of last $$n$$ terms. Then $$S_1, S_2, S_3$$ are in
A.
A.P.
B.
G.P.
C.
H.P.
D.
None of these
Answer :
G.P.
Solution :
Let the $$3n$$ terms of G.P. be
$$\eqalign{
& a,ar,a{r^2},.....\,a{r^{n - 1}},a{r^n},a{r^{n + 1}},.....\,a{r^{2n - 1}},a{r^{2n}},{a^{2n + 1}},.....,a{r^{3n - 1}}.\,\,{\text{Then}} \cr
& {S_1} = a + ar + a{r^2} + ..... + a{r^{n - 1}} = \frac{{a\left( {1 - {r^n}} \right)}}{{1 - r}} \cr
& {S_2} = a{r^n} + a{r^{n + 1}} + ..... + a{r^{2n - 1}} = \frac{{a{r^n}\left( {1 - {r^n}} \right)}}{{1 - r}} \cr
& {S_3} = a{r^{2n}} + a{r^{2n + 1}} + ..... + a{r^{3n - 1}} = \frac{{a{r^{2n}}\left( {1 - {r^n}} \right)}}{{1 - r}} \cr
& {\text{Clearly, }}\frac{{{S_2}}}{{{S_1}}} = \frac{{{S_3}}}{{{S_2}}} = {r^n} \cr} $$