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Q.
The third term of a $G.P$. is $42$. Find the product of its first five terms.
Sequences and Series
Solution:
Let $a$ be the first term and $r$ be the common ratio. Then, $a_{3} = 42$
$\Rightarrow ar^{2 }= 42 \quad...\left(i\right)$
$\therefore $ Product of first five terms $= a_{1}a_{2}a_{3}a_{4}a_{5}$
$ = a\left(ar\right)\left(ar^{2}\right)\left(ar^{3}\right)\left(ar^{4}\right) $
$= a^{5}r^{10}$
$ = \left(ar^{2}\right)^{5} $
$= \left(42\right)^{5} $ [Using $(i)$]