Obtain the nth term and sum of the following terms upto n terms

(a) S_n = 1.2.3 + 3.4.5 + 5.6.7 + 7.8.9+.........

(b) S_n = 1.2.3.4 + 2.3.4.5 + 3.4.5.6 + ............

1+2+3+4+...........+n      =  n²/2 + n/2

1²+2²+3²+ ....... +n²     = n³/3 + n²/2 + n/6

1³+2³+3³+ ....... +n3     = n⁴/4 + n³/2 + n²/4

1⁴+2⁴+3⁴+ ....... +n⁴     = n⁵/5 + n⁴/2 + n³/3 - n/30

a) S_n = 1.2.3 + 3.4.5 + 5.6.7 + 7.8.9+.........

n is set of natural no's 1,2,3.......n
n th term is (2n-1).(2n).(2n+1)

sum of n terms

               = 8( sigma: n³ ) - 2( sigma : n )

               = 8.n²(n+1)²/4 -n(n+1)

               = n(n+1).(2n²+2n-1)

b) S_n = 1.2.3.4 + 2.3.4.5 + 3.4.5.6 + ............

n is set of natural no's 1,2,3.......n
n th term is: n(n+1)(n+2)(n+3)

         = n⁴ + 6n³+ 11n² + 6n

sum of n terms

               = ( sigma: n⁴ ) + 6( sigma : n³ ) + 11.( sigma: n² ) + 6( sigma : n )

               =  (n⁵/5 + n⁴/2 + n³/3 - n/30) +6(n⁴/4 + n³/2 + n²/4)+11(n³/3 + n²/2 + n/6)+6(n²/2 + n/2)              

               =  n⁵/5 + 2.n⁴ + 7.n³ +10. n²+  9n/2

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