Question

Simplify
5*25^n+1-25*5^2n / 5*5^2n+3-25^n+1

Answer 1

Answer:

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Answer 2

Given expression is :-

$\sf \purple{ :\longmapsto\:\dfrac{5 \times {25}^{n + 1} - 25 \times {5}^{2n} }{5 \times {5}^{2n + 3} - {25}^{n + 1} } }$

Can be rewritten as :-

$\sf\:  =  \: \dfrac{5 \times { {(5}^{2} )}^{n + 1} - {5}^{2} \times {5}^{2n} }{5 \times {5}^{2n + 3} - {( {5}^{2} )}^{n + 1} }$

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$\purple{\sf \:\boxed{\sf{ {( {x}^{m} )}^{n} \: = \: {x}^{mn}}}}$

$\purple{\sf \:\boxed{\sf{ \: \: {x}^{m} \times {x}^{n} = {x}^{m + n} \: }}}$

So, using this identity, we :-

$\sf \:  =  \: \dfrac{5 \times {5}^{2n + 2} - {5}^{2n + 2} }{{5}^{2n + 3 + 1} - {5}^{2n + 2} }$

$\sf \:  =  \: \dfrac{{5}^{2n + 2 + 1} - {5}^{2n + 2} }{{5}^{2n + 4} - {5}^{2n + 2} }$

$\sf \:  =  \: \dfrac{{5}^{2n + 2 + 1} - {5}^{2n + 2} }{{5}^{2n + 2 + 2} - {5}^{2n + 2} }$

$\sf \:  =  \: \dfrac{ {5}^{2n + 2} (5 - 1)}{ {5}^{2n + 2} ( {5}^{2} - 1)}$

$\sf\:  =  \: \dfrac{4}{25 - 1}$

$\sf \:  =  \: \dfrac{4}{24}$

$\sf\purple{ \:  =  \: \dfrac{1}{6}}$

Hence–

$\purple{\bf :\longmapsto\:\boxed{\sf{ \dfrac{5 \times {25}^{n + 1} - 25 \times {5}^{2n} }{5 \times {5}^{2n + 3} - {25}^{n + 1} } = \frac{1}{6} }}}$

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