Prove that `7^n(1+n/7+(n(n-1))/(7.14)+(n(n-1)(n-2))/(7.14.21)...)=4^n(1+n/2+(n(n+1))/(2.4)+(n(n+1)(n+2))/(2.4.6)....)`

Prove that 5^(n) (1+(n)/(5) +(n(n-1))/(5*10) +(n(n-1)(n-2))/(5*10*15)+…oo)=3^(n) (1+(n)/(2)+(n(n+1))/(2*4)+(n(n+1)(n+2))/(2*4*6)+…oo)

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

Prove that 1*2+2*3+3*4+.....+n*(n+1)=(n(n+1)(n+2))/(3)

Prove that (1)/(n!)+(1)/(2!(n-2)!)+(1)/(4!(n-4)!)+...=(1)/(n!)2^(n-1)

Prove that 1^2/(1.3)+2^2/(3.5)+3^2/(5.7)+.....+n^2/((2n-1)(2n+1))=((n)(n+1))/((2(2n+1))

Prove that 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)

By the principle of mathematical induction prove that the following statements are true for all natural numbers 'n' (a) (1)/(1.3)+(1)/(3.5)+(1)/(5.7)+......+(1)/((2n-1)(2n+1)) =(n)/(2n+1) (b) (1)/(1.4)+(1)/(4.7)+(1)/(7.10)+......+(1)/((3n-2)(3n+1)) =(n)/(3n+1)

(2.3^(n+1)+7.3^(n-1))/(3^(n+1)-2((1)/(3))^(1-n))=