Convert the following products into factorial: `(n+1)(n+2)(n+3).......(2n)`

Convert the following products into factorials: (iii) (n + 1) (n +2) (n + 3).... (2n) (iv) 1.3.5 . 7 . 9.... (2n-1)

Convert the following products into factorial: 1. 3. 5. 7. 9 (2n-1)

Prove the following by using induction for all n in N . 1+2+3+.....+n=(n(n+1))/(2)

Find the sum of the following series to n term: (n^2-1^2)+2(n^2-2^2)+3(n^2-3^2)+……...

Prove that ((2n+1)!)/(n !)=2^n{1. 3. 5 .........(2n-1)(2n+1)}

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

Prove the following by the principle of mathematical induction: \ 7^(2n)+2^(3n-3). 3^(n-1) is divisible 25 for all n in Ndot

Prove the following by the principle of mathematical induction: \ 1. 2+2. 3+3. 4++n(n+1)=(n(n+1)(n+2))/3

Prove the following by the principle of mathematical induction: 1/(3. 5)+1/(5. 7)+1/(7. 9)+1/((2n+1)(2n+3))=n/(3(2n+3))

By the principle of mathematical induction prove that for every n in N, the following statements are true: 1 +5 +9 + ....+ (4n - 3) = 2n^2 - n