For all `n ge 1` prove that
`(1)/(1.2)+ (1)/(2.3)+(1)/(3.4)+.....+(1)/(n(n+1))=(n)/(n+1)`

Prove that 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))

lim_(n rarr oo) ((1)/(1.2) + (1)/(2.3) + (1)/(3.4) +…..+ (1)/(n(n+1))) is :

Prove that by using the principle of mathematical induction for all n in N : (1)/(1.4)+ (1)/(4.7)+(1)/(7.10)+...+ (1)/((3n-2)(3n+1))= (n)/(3n+1)

Prove that by using the principle of mathematical induction for all n in N : (1)/(3.5)+ (1)/(5.7)+ (1)/(7.9)+ ....+(1)/((2n+1)(2n+3))= (n)/(3(2n+3))

Prove that by using the principle of mathematical induction for all n in N : (1)/(2.5)+ (1)/(5.8) + (1)/(8.11)+ ...+(1)/((3n-1)(3n+2))= (n)/(6n+4)

Prove that by using the principle of mathematical induction for all n in N : (1)/(2)+ (1)/(4)+ (1)/(8)+ ......+ (1)/(2^(n))= 1-(1)/(2^(n))

For all n ge 1 prove that 1^(2)+2^(2)+ 3^(2)+4^(2)+….+n^(2)= (n(n+1)(2n+1))/(6)

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

For a positive integer n, let a_(n) = 1+( 1)/( 2) +( 1)/( 3) + ( 1)/( 4) + "......." + ( 1)/(( 2^(n) - 1)) . Then :

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