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Mathematics

Simplify :

5×(25)n+1−25×52n5×5(2n+3)−(25)n+1\dfrac{5 \times (25)^{n + 1} - 25 \times 5^{2n}}{5 \times 5^{(2n + 3)} - (25)^{n + 1}}

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Answer

Given,

5×(25)n+1−25×52n5×5(2n+3)−(25)n+1\dfrac{5 \times (25)^{n + 1} - 25 \times 5^{2n}}{5 \times 5^{(2n + 3)} - (25)^{n + 1}}

Simplifying the expression :

⇒5×(25)n+1−52×52n51×52n+3−(25)n+1⇒5×[(5)2]n+1−52n+252n+3+1−[(5)2]n+1⇒51×(5)2n+2−52n+252n+4−(5)2n+2⇒52n+2+1−52n+252n+4−52n+2⇒52n+3−52n+252n+4−52n+2⇒52n+2(5−1)52n+2(52−1)⇒4(25−1)⇒424⇒16.\Rightarrow \dfrac{5 \times (25)^{n + 1} - 5^2 \times 5^{2n}}{5^1 \times 5^{2n + 3} - (25)^{n + 1}} \\[1em] \Rightarrow \dfrac{5 \times [(5)^2]^{n + 1} - 5^{2n + 2}}{5^{2n + 3 + 1} - [(5)^2]^{n + 1}} \\[1em] \Rightarrow \dfrac{5^1 \times (5)^{2n + 2} - 5^{2n + 2}}{5^{2n + 4} - (5)^{2n + 2}} \\[1em] \Rightarrow \dfrac{5^{2n + 2 + 1} - 5^{2n + 2}}{5^{2n + 4} - 5^{2n + 2}} \\[1em] \Rightarrow \dfrac{5^{2n + 3} - 5^{2n + 2}}{5^{2n + 4} - 5^{2n + 2}} \\[1em] \Rightarrow \dfrac{5^{2n + 2}(5 - 1)}{5^{2n + 2}(5^2 - 1)} \\[1em] \Rightarrow \dfrac{4}{(25 - 1)} \\[1em] \Rightarrow \dfrac{4}{24} \\[1em] \Rightarrow \dfrac{1}{6}.

Hence, 5×(25)n+1−25×52n5×52n+3−(25)n+1=16\dfrac{5 \times (25)^{n + 1} - 25 \times 5^{2n}}{5 \times 5^{2n + 3} - (25)^{n + 1}} = \dfrac{1}{6}.

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