Question

the third term of G.P is 10 then the product of first five term is​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

The third term of a Geometric Progression is 10

TO DETERMINE

The product of first five terms of the progression

CALCULATION

Let the first five terms of the Geometric Progression be

$\displaystyle \sf{ \frac{a}{ {r}^{2} } \: , \frac{a}{r} \: , \: a \: , \: ar \:, \: a {r}^{2} \: }$

Now it is given that the third term = a = 10

Hence the product of first five terms of the progression is

$= \displaystyle \sf{ \frac{a}{ {r}^{2} } \: \times \frac{a}{r} \: \times \: a \: \times \: ar \: \times \: a {r}^{2} \: }$

$\sf{ = {a}^{5} \: }$

$= \sf{ {10}^{5} \: }$

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Answer 2

$Let \: 'a'\: and \: 'r' \: are \: first \:term \:and \\common \:ratio \: of \: a \: G.P$

$Third \: term (a_{3}) = 10 \: (given)$

$\boxed{ \pink{ n^{th} \:term (a_{n}) = ar^{n-1} }}$

$\implies ar^{2} = 10 \: --(1)$

$\red{ Product \: of\: first \: 5 \: terms }$

$= a \times ar \times ar^{2} \times ar^{3} \times ar^{4}$

$= a^{5} \times r^{10}$

$= (ar^{2})^{5}$

$= 10^{5} \: [ From \: (1) ]$

Therefore.,

$\red{ Product \: of\: first \: 5 \: terms }\green { = 10^{5}}$

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