Question

Find the sum to n terms of the series whose nth term is given by n(n+1)(n+4)

Answer 1

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Answer 2

Answer:

StTn = n(n + 1)( n + 4)

we know,

Sn = ∑Tn

=∑{n(n + 1)(n + 4)}

=∑{n³ + 5n² + 4n }

=∑n³ + 5∑n² + 4∑n

we know ,

∑n³ = [n(n + 1)/2]²

∑n² = n(n + 1)(2n + 1)/6

∑n = n( n + 1)/2 , use this here,

Sn = [n(n + 1)/2]² + 5n(n +1)(2n+1)/6 + 4n(n+1)/2

= n(n +1)/2[ n(n+1)/2 + 5(2n +1)/3 + 4]

= n(n + 1)/2[ {3n² + 3n + 20n + 10 + 24}/6]

= n(n+1)/12[3n² + 23n + 34]

= n(n+1)/12 [ 3n² + 6n + 17n + 34]

= n(n +1)/12 (3n + 17)(n+2)

= n(n+1)(n+2)(3n+17)/12