1+2+3+...+n = (1/2)n(n+1) = Tn

1^2+2^2+...+n^2= (1/6)n(n+1)(2n+1) = Rn

an

= n^3

=(n-1)n(n+1) +n

= (1/4) [(n-1)n(n+1)(n+2)- (n-2)(n-1)n(n+1) ] + (1/2)[ n(n+1) - (n-1)n ]

Pn

= a1+a2+a3+...+an

=(1/4)(n-1)n(n+1)(n+2)+ (1/2)n(n+1)

=(1/4)n(n+1)[(n-1)(n+2) + 2 ]

=(1/4)[n(n+1)]^2

ie

1^3+2^3+....+n^3 = (1/4)[n(n+1)]^2

bn

=n^4

=(n-1)n(n+1)(n+2) - 2n^3 +n^2 +2n

=(1/5)[(n-1)n(n+1)(n+2)(n+3) -(n-2)(n-1)n(n+1)(n+2)] - 2n^3 +n^2 +2n

Sn

=b1+b2+...+bn

=(1/5)(n-1)n(n+1)(n+2)(n+3) -2Pn +Rn +2Tn

=(1/5)(n-1)n(n+1)(n+2)(n+3) -(1/2)[n(n+1)]^2

+(1/6)n(n+1)(2n+1) + n(n+1)

=(1/30)n(n+1) [ 6(n-1)(n+2)(n+3) - 15n(n+1) + 5(2n+1) + 30]

=(1/30)n(n+1) [ 6(n^3+4n^2+n-6) - 15(n^2+n) + 5(2n+1) + 30]

=(1/30)n(n+1) [ (6n^3+24n^2+6n-36) - (15n^2+15n) + (10n+5) + 30]

=(1/30)n(n+1)(6n^3+9n^2+n-1)

ie

1^4+2^4+...+n^4

=(1/30)n(n+1)(6n^3+9n^2+n-1)