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Question

State true or false.
13+33+53+...+(2n−1)3=n2(2n2−1) n is a natural number

A
True
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B
False
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Solution

The correct option is A True
13+33+53+...+(2n1)3=2n4n2

The result is true for n=1
2n4n2=2(1)4(1)2=21=1

Let the result be true for n=k. That is
13+33+53+...+(2k1)3=2k4k2

Now we need to prove that the result is also true for n=k+1. That is
13+33+53+...+(2k1)3+(2(k+1)1)3=2k4k2+(2(k+1)1)
=2k4k2+(2k+1)3
=2k4k2+8k3+3×4k2+3×2k+1
=2k4+8k3+12k2+8k+22k1k2
=2(k4+4k3+6k2+4k+1)(k2+2k+1)
=2(k+1)4(k+1)2

The result is also true for n=k+1.

Hence by the principle of mathematical induction the result is true for all nN

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