Question

Докажите тождество:
$frac{1}{log_{a}k} + frac{1}{log_{a^2}k} + frac{1}{log_{a^3}k} + frac{1}{log_{a^4}k}+ frac{1}{log_{a^5}k} =15log_{k}{a}$

Answer 1

$dfrac{1}{log_{a}k} + dfrac{1}{log_{a^2}k} + dfrac{1}{log_{a^3}k} + dfrac{1}{log_{a^4}k}+ dfrac{1}{log_{a^5}k} =15log_{k}{a}$

Преобразовываем левую часть к правой:
$dfrac{1}{log_{a}k} + dfrac{1}{log_{a^2}k} + dfrac{1}{log_{a^3}k} + dfrac{1}{log_{a^4}k}+ dfrac{1}{log_{a^5}k} = \ = dfrac{1}{log_{a}k} + dfrac{1}{ frac{1}{2} log_{a}k} + dfrac{1}{ frac{1}{3}log_{a}k} + dfrac{1}{ frac{1}{4}log_{a}k}+ dfrac{1}{ frac{1}{5}log_{a}k} = \ = dfrac{1}{log_{a}k} + dfrac{2}{ log_{a}k} + dfrac{3}{ log_{a}k} + dfrac{4}{ log_{a}k}+ dfrac{5}{ log_{a}k} = \ = dfrac{15}{ log_{a}k}= dfrac{15}{ frac{1}{log_{k}a} }=15log_{k}a$