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Question

Prove the following by using the principle of mathematical induction for all nN:12+14+18+.....+12n=112n

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Solution

Let the given statement be P(n), i.e.,
P(n):12+14+18+.....+12n=112n
For n=1, we have
P(1)=12=1121=12, which is true.
Let P(k) be true for some kN, i.e.,
12+14+18+.....+12k=112k......(i)
We shall now prove that P(k+1) is true.
Consider (12+14+18+.....+12k)+12k+1
=(112k)+12k+1
=112k+12.2k
=112k(112)
=112k(12)
=112k+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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