Question

solve it please?????​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{n=2}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies {25}^{n - 1} + 100 = {5}^{2n - 1} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies n = ?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {25}^{n - 1} + 100 = {5}^{2n - 1} \\ \\ \tt: \implies 100 = {5}^{2n - 1} - {25}^{n - 1} \\ \\ \tt: \implies {5}^{2n - 1} - {25}^{n - 1} = 100 \\ \\ \tt: \implies {5}^{2n} \times {5}^{ - 1} - {25}^{n} \times {25}^{ - 1} = 100 \\ \\ \tt: \implies ({5}^{2} ) ^{n} \times \frac{1}{5} - {25}^{n} \times \frac{1}{25} = 100 \\ \\ \tt: \implies \frac{ {25}^{n} }{5} - \frac{ {25}^{n} }{25} = 100 \\ \\ \tt: \implies \frac{5 \times {25}^{n} - {25}^{n} }{25} = 100 \\ \\ \tt: \implies {25}^{n} (5 - 1) = 100 \times 25 \\ \\ \tt: \implies {25}^{n} \times 4 =2500 \\ \\ \tt: \implies {25}^{n} = \frac{2500}{4} \\ \\ \tt: \implies {25}^{n} = 625 \\ \\ \tt: \implies {25}^{n} = {25}^{2} \\ \\ \green{\tt: \implies n = 2}$

Answer 2

Given :-

$\tt{25^n^-^1+100=5^2^n^-^1}$

To find :-

The value of 'n'.

Solution :-

$\displaystyle{\tt{5^{2n-2}+100=5^{2n-1}}}\\\\\displaystyle{\tt{\implies \frac{5^{2n-1}}{5}+100=5^{2n-1}} }\\\\\displaystyle{\texttt{Let }}5^{2n-1}\texttt{ be x.}\\\\\displaystyle{\tt{\implies \frac{x}{5}+100=x }}\\\\\displaystyle{\tt{\implies 100=x-\frac{x}{5} }}\\\\\displaystyle{\tt{\implies 100=\frac{4x}{5} }}\\\\\displaystyle{\tt{\implies x=\frac{500}{4}=125 }}$

$\tt{Now,}\\\\\displaystyle{\tt{5^{2n-1}=125}}\\\\\displaystyle{\tt{\implies 5^{2n-1}=5^3}}\\\\\displaystyle{\tt{\implies 2n-1=3}}\\\\\tt{As\ the\ base\ is\ same\ in\ both\ the\ sides\ i.e.\ 5.}\\\\\displaystyle{\tt{n=\frac{3+1}{2} }}\\\displaystyle{\tt{\boxed{\implies n=2}}}$