`1/4.5 + 1/5.6 + .......+ 1/((n+3)(n+4)`

Using the mathematical induction, show that for any natural number n, 1/2.5 + 1/5.8 + 1/8.11 + …+ 1/((3n-1)(3n+2))=n/(6n+4)

Lt_(n to oo) ((1)/(1.5)+(1)/(5.9)+....+(1)/((4n-3)(4n+1)))=

lim_ (n rarr oo) ((1 + 2 ^ (4) + 3 ^ (4) + ...... + n ^ (4)) / (n ^ (5))) - lim_ (n rarr oo ) ((1 + 2 ^ (6) + 3 ^ (6) + .... + n ^ (6)) / (n ^ (7)))

Using mathimatical induction prove that 1/(2.5)+1/(5.8)+1/(8.11)+......+1/((3n-1)(3n+2))=n/(6n+4) for all n in N .

(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)

1/(2*5)+1/(5*8)+1/(8*11)+............1/((3n-1)(3n+2))=n/((6n+4)) forall n in N.

The sum of n terms of the series : (1)/( 1.2.3.4) + ( 1)/( 2.3.4.5) + ( 1) / ( 3.4.5.6.)+"......." is :

(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))