`1.4+2.5+........+ n(n+3) =`

By method of induction prove that 1.3 + 2.5 + 3.7 +...+ n (2n + 1) = n/6 (n + 1) (4n + 5) for all n in N

Find sum the series (n)/(1.2*3)+(n-1)/(2.3*4)+(n-2)/(3.4*5)+......... up to n terms..-

1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^(2)+6n-1))/(3)

1.3 + 2.4 + 3.5 + …… + n(n+2) =( n(n+1)(2n+7))/6

Show that (.^(4n)C_(2n))/(.^(2n)C_(n))=(1.3.5......(4n-1))/({1.3.5......(2n-1)}^(2)) .

1.2.3+2.3.4+....+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4

By the definition of the definite integral, the value of lim_(ntooo)((1^(4))/(1^(5)+n^(5))+(2^(4))/(2^(5)+n^(5))+(3^(4))/(3^(5)+n^(5))+............+(n^(4))/(n^(5)+n^(5)))