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8. Sequences and Series
easy
If the sum of three terms of $G.P.$ is $19$ and product is $216$, then the common ratio of the series is
A
$ - \frac{3}{2}$
B
$\frac{3}{2}$
C
$2$
D
$3$
Solution
(b) Let three terms of $G.P.$ are $a,\;ar,\;a{r^2}$. Then
$a + ar + a{r^2} = 19 $
$\Rightarrow a[1 + r + {r^2}] = 19$ …..$(i)$
$a\;.\;ar\;.\;a{r^2} = 216$
$\Rightarrow {a^3}{r^3} = 216$
$\Rightarrow ar = 6$ …..$(ii)$
Dividing $(ii)$ by $(i),$
$\frac{6}{r} + \frac{6}{r}r + \frac{6}{r}{r^2} = 19 $
$\Rightarrow \frac{6}{r} + 6 + 6r = 19$
$ \Rightarrow {r^2} – \frac{{13}}{6}r + 1 = 0$.
Hence $r = \frac{3}{2}$.
Standard 11
Mathematics
