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  4. If If n =(2020), then (1/ log2n)+ (1/ log3n)+ (1/ log4n)+............+ (1/ log2020 n)

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Q. If If $n =(2020)$, then $\frac {1}{\log_2n}+\frac {1}{\log_3n}+\frac {1}{\log_4n}+............+\frac {1}{\log_{2020} n}$

KCETKCET 2009Sequences and Series

Solution:

$\frac{1}{\log _{2} n}+\frac{1}{\log _{3} n}+\frac{1}{\log _{4} n}+\ldots+\frac{1}{\log _{2020} n}$
$=\log _{n} 2+\log _{n} 3+\log _{n} 4+\ldots+\log _{n} 2020$
$=\log _{n}(2 \times 3 \times 4 \times \ldots \times 2020)$
$=\log _{(2020) 1}(2020) ! \,\,\,\,\, (\because n=2020 !$ given $)$
$=1$