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Simplify: ( 5×(25)^(n+1) - 25 × (5)^(2n))/(5×(5)^(2n+3 )- (25)^(n+1​)) \\

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User Sookie
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\green{\large\underline{\sf{Solution-}}}

Given expression is


\rm :\longmapsto\:\frac{5 * {25}^(n + 1) - 25 * {5}^(2n) }{5 * {5}^(2n + 3) - {25}^(n + 1) }

can be rewritten as


\rm \:  =  \: \frac{5 * { {(5}^(2) )}^(n + 1) - {5}^(2) * {5}^(2n) }{5 * {5}^(2n + 3) - {( {5}^(2) )}^(n + 1) }

We know,


\purple{\rm :\longmapsto\:\boxed{\tt{ {( {x}^(m) )}^(n) \: = \: {x}^(mn)}}} \\

And


\purple{\rm :\longmapsto\:\boxed{\tt{ \: \: {x}^(m) * {x}^(n) = {x}^(m + n) \: }}} \\

So, using this identity, we


\rm \:  =  \: \frac{5 * {5}^(2n + 2) - {5}^(2n + 2) }{{5}^(2n + 3 + 1) - {5}^(2n + 2) }


\rm \:  =  \: \frac{{5}^(2n + 2 + 1) - {5}^(2n + 2) }{{5}^(2n + 4) - {5}^(2n + 2) }

can be further rewritten as


\rm \:  =  \: \frac{{5}^(2n + 2 + 1) - {5}^(2n + 2) }{{5}^(2n + 2 + 2) - {5}^(2n + 2) }


\rm \:  =  \: \frac{ {5}^(2n + 2) (5 - 1)}{ {5}^(2n + 2) ( {5}^(2) - 1)}


\rm \:  =  \: (4)/(25 - 1)


\rm \:  =  \: (4)/(24)


\rm \:  =  \: (1)/(6)

Hence,


\rm :\longmapsto\:\boxed{\tt{ \frac{5 * {25}^(n + 1) - 25 * {5}^(2n) }{5 * {5}^(2n + 3) - {25}^(n + 1) } = (1)/(6) }}

User Don Grem
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