`(1^4)/1.3+(2^4)/3.5+(3^4)/5.7+......+n^4/((2n-1)(2n+1))=(n(4n^2+6n+5))/48+n/(16(2n+1)`

Statement -1: (1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+ . . . .+(n^(2))/((2n-1)(2n+1))=(n(n+1))/(2(2n+1)) Statement -2: (1)/(1.3)+(1)/(3.5)+(1)/(5.7)+ . . . .+(1)/((2n-1)(2n+1))=(1)/(2n+1)

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

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)

Find the sum (1^4)/(1xx3)+(2^4)/(3xx5)+(3^4)/(5xx7)+......+(n^4)/((2n-1)(2n+1))

Using mathematical induction, prove that (1)/(1.3.5) + (2)/(3.5.7) +….+(n)/((2n-1)( 2n+1) ( 2n+3)) =( n(n+1))/( 2(2n+1) (2n+3))

Prove the following by using the principle of mathematical induction for all n in N : 1/(1. 2. 3)+1/(2. 3. 4)+1/(3. 4. 5)+...+1/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2))

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + … + C_(n) x^(n) , Show that (2^(2))/(1*2) C_(0) + (2^(3))/(2*3) C_(1) + (2^(4))/(3*4)C_(2) + ...+ (2^(n+2)C_n)/((n+1)(n+2))= (3^(n+2)-2n-5)/((n+1)(n+2))

Using the principle of mathematical induction prove that 1/(1. 2. 3)+1/(2. 3. 4)+1/(3. 4. 5)++1/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2) for all n in N

lim_(n->oo) {1/1.3+1/3.5+1/5.7+.....+1/((2n+1)(2n+3)) is equal to

Prove that 1/(n+1)=(.^n C_1)/2-(2(.^n C_2))/3+(3(.^n C_3))/4- . . . +(-1^(n+1))(n*(.^n C_n))/(n+1) .