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

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

Find the value of sum_(i=1)^n sum_(i=1)^n sum_(k=1)^n k

The value of the positive integer n for which the quadratic equation sum_(k=1)^n(x+k-1)(x+k)=10n has solutions alpha and alpha+1 for some alpha is

The value of the positve integer n for which the quadratic equation sum_(k=1)^n(x+k-1)(x+k)=10n has solutions alpha and alpha+1 for some alpha is

The value of the positve integer n for which the quadratic equation sum_(k=1)^(n)(x+k-1)(x+k)=10n has solutions alpha and alpha+1 for some alpha is

The value of the positve integer n for which the quadratic equation sum_(k=1)^n(x+k-1)(x+k)=10n has solutions alpha and alpha+1 for some alpha is

The value of the positve integer n for which the quadratic equation sum_(k=1)^n(x+k-1)(x+k)=10n has solutions alpha and alpha+1 for some alpha is

Evaluate : sum_(k=1)^n (2^k+3^(k-1))

Evaluate : sum_(k=1)^n (2^k+3^(k-1))