# Obtain the nth term and sum of the following terms upto n terms(a) Sn = 1.2.3 + 3.4.5 + 5.6.7 + 7.8.9+.........(b) Sn = 1.2.3.4 + 2.3.4.5 + 3.4.5.6 + ............

Ramesh V
70 Points
14 years ago

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) Sn = 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) Sn = 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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Naga Ramesh
IIT Kgp - 2005 batch