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Question

Prove the following by using the principle of mathematical induction for all nN:12.5+15.8+18.11+......+1(3n1)(3n+2)=n(6n+4)

Solution
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Let the given statement be P(n) i.e
P(n):12.5+15.8+18.11+......+1(3n1)(3n+2)=n(6n+4)
For n=1, we have
P(1)=12.5=110=16.1+4+110, which is true.
Let P(k) be true for some kN i.e.,
12.5+15.8+18.11+......+1(3k1)(3k+2)=k6k+4.......(i)
We shall now prove that P(k+1) is true.
Consider
12.5+15.8+18.11+.......+1(3k1)(3k+2)+1{3(k+1)1}{3(k+1)+2}
=k6k+4+1(3k+31)(3k+3+2
=k6k+4+1(3k+2)(3k+5)
=1(3k+2)(k2+13k+5)
=1(3k+2)(k(3k+5)+22(3k+5))
=1(3k+2)((3k+2)(k+1)2(3k+5))
=(k+1)6k+10
=(k+1)6(k+1)+4
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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