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SEQUENCES AND SERIES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSEQUENCES AND SERIES3 Marks
Question
The Sum of first three terms of a G.P. is $16$ and the Sum of the next three term is $128$. determine the first term, the common ratio and the Sum to n terms of the G.P.
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Answer

$S_3=16$
$\frac{a\left(1-r^3\right)}{1-r}=16(1)$
$S_6-S_3=128$
$\frac{a\left(1-r^6\right)}{1-r}-16=128$
$\frac{a\left(1-r^6\right)}{1-r}=144(2)$
$(2) \div(1)$
$\frac{1-r^6}{1-r^3}=\frac{144}{16}$
$1+\mathrm{r}^3=9$
$\mathrm{r}^3=8$
$\mathrm{r}=2$
$s_3=\frac{a\left(r^3-1\right)}{r-1}=16$
$\mathrm{a}=16 / 7$
$s_n=\frac{a\left(r^n-1\right)}{r-1}=\frac{16}{7} \frac{\left(2^n-1\right)}{2-1}=\frac{16}{7}\left(2^n-1\right)$

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