Question

find the geometric mean of the first 25 powers of 25​

Answer 1

$\textbf{Given:}$

$\text{Numbers 25^1,25^2,25^3,.........25^{25} }$

$\textbf{To find:}$

$\text{Geometric mean of given numbers}$

$\textbf{Solution:}$

$\text{We know that,}$

$\text{Geometric mean of a_1,a_2,a_3,.........,a_n is}$

$\bf\displaystyle\sqrt[n]{a_1{\times}a_2{\times}a_3{\times}.........{\times}a_n}$

$\textbf{Geometric mean of first 25 powers of 25}$

$=\displaystyle\sqrt[25]{25^1{\times}25^2{\times}25^3{\times}.........{\times}25^{25}}$

$=\displaystyle\sqrt[25]{25^{(1+2+3+..........+25)}}$

$=\displaystyle\sqrt[25]{25^{\frac{25{\times}26}{2}}}$

$=\displaystyle\sqrt[25]{25^{25{\times}13}}$

$=\displaystyle(25^{25{\times}13})^{\frac{1}{25}}$

$=\displaystyle25^{13}$

$\therefore\textbf{Geometric mean of first 25 powers of 25 is \displaystyle\bf25^{13} }$

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