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8. Sequences and Series
hard
The sum of the $3^{rd}$ and the $4^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000$. If the first term of this $G.P.$ is positive, then its $7^{th}$ term is
A
$7290$
B
$640$
C
$2430$
D
$320$
(JEE MAIN-2015)
Solution
Let $a, ar$ and $a{r^2}$ be the first three terms of $G.P$
According to the question
$a\left( {ar} \right)\left( {a{r^2}} \right) = 1000 \Rightarrow {\left( {ar} \right)^3} = 1000 \Rightarrow ar = 10$
and $a{r^2} + a{r^3} = 60 \Rightarrow ar\left( {r + {r^2}} \right) = 60$
$ \Rightarrow {r^2} + r – 6 = 0$
$ \Rightarrow r = 2, – 3$
$a = 5,a = – \frac{{10}}{3}$ (reject)
Hence, ${T_7} = a{r^6} = 5{\left( 2 \right)^6} = 5 \times 64 = 320$
Standard 11
Mathematics
