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Question

Using the principle of mathematical induction, prove the following for all nN:
11.4+14.7+17.10+...+1(3n2)(3n+1)=n(3n+1)

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Solution

Let P(n):11.4+14.7+17.10+....+1(3n2)(3n+1)=n(3n+1)
For n=1,LHS=11.4=14
RHS=13+1=14
LHS=RHS
P(1) is true
Lte us assume P(k) is true for some kN
i.e, 11.4+14.7+17.10+....+1(3n2)(3k2)=n(3k+1)
=k3k+1
Adding (k+1)th term =1(2k+1)(3k+4)
=k3k+1
Adding (k+1)th term =1(2k+1)(3k+4)
on both sides, we get
11.4+14.7+17.10+....+1(3k1)(3k+1)+1(3k+1)(3k+4)
=k3k+1+1(3k+1)(3k+4)
=k(3k+4)+1(3k1)(3k+4)=3k2+4k+1(3k+1)(3k+4)
=k+13(k+1)+1
which is P(k+1)
Thus, P(k)P(k+1)
Hence, by mathematical induction P(n) is true for all nN.

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