For any integer `n>=1`, then `sum_(k=1)^n k(k+2)=`

For any integer n ge 1 , then sum_(k=1)^(n) k ( k+2) is equal to

If a_(k) = (1)/( k(k+1) ) for k= 1,2,3,….n then (sum_(k=1)^(n) a_(k) )=

For which positive integers n is the ratio (sum_(k=1)^(n)k^(2))/(sum_(k=1)^(n)k) an integer?

For which positive integers n is the ratio, (sum+(k=1)^(n) k^(2))/(sum_(k=1)^(n) k) an integer ?

Suppose det [{:(sum_(k=0)^(n)k,,sum_(k=0)^(n).^nC_(k)k^2),(sum_(k=0)^(n).^nC_(k)k,,sum_(k=0)^(n).^nC_(k)3^(k)):}]=0 holds for some positive integer n. then sum_(k=0)^(n)(.^nC_(k))/(k+1) equals ............