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Question

Prove the following by using the principle of mathematical induction for all nN:11.4+14.7+17.10+.....+1(3n2)(3n+1)=n(3n+1)

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Solution

Let the given statement be P(n) i.e.,
P(n):11.4+14.7+17.10+.....+1(3n2)(3n+1)=n(3n+1)
For n=1, we have
P(1)=11.4=13.1+1=14=11.4, which is true.
Let P(k) be true for some kN, i.e.,
P(k)=11.4+14.7+17.10+.....+1(3k2)(3k+1)=k(3k+1)......(i)
We shall now prove that P(k+1) is true.
Consider
11.4+14.7+17.10+.....+1(3k2)(3k+1)+1{3(k+1)2}{3(k+1)+1} [Using (i)]
=k3k+1+1(3k+1)(3k+4)
=1(3k+1){k+1(3k+4)}
=1(3k+1){k(3k+4)+1(3k+4)}
=1(3k+1){3k2+4k+1(3k+4)}
=(3k+1)(k+1)(3k+1)(3k+4)
=(k+1)3(k+1)+1
Thus P(k+1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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