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8. Sequences and Series
hard
${{(0.2)}^{{{\log }_{\sqrt{5}}}\left( \frac{\text{1}}{\text{4}}\,+\,\frac{\text{1}}{\text{8}}\,+\,\frac{\text{1}}{\text{16}}\,+\,.....\,\infty \right)}}$ નું મૂલ્ય:
A
$1$
B
$2$
C
$1/2$
D
$4$
Solution
અહીં $\frac{\text{1}}{\text{4}}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,…..\,\infty \,$
$=\,\frac{\frac{1}{4}}{1-\frac{1}{2}}\,=\,\frac{1}{2}\,=\,{{2}^{-1}}$
$\therefore \,{{(0.2)}^{{{\log }_{\sqrt{5}}}\,\left( \frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,…\,\infty \right)}}$
$\,=\,{{\left( \frac{1}{5} \right)}^{{{\log }_{\sqrt{5}}}2^{-1}}}$
$=\,{{\left( {{5}^{-1}} \right)}^{\frac{-1}{1/2}{{\log }_{5}}2}}$
$=\,\,{{5}^{2{{\log }_{5}}2}}\,\,\,=\,{{5}^{{{\log }_{5}}{{(2)}^{2}}}}\,=\,{{2}^{2}}\,\,=\,4$
Standard 11
Mathematics
