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Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: 1.2 + 2.22 + 3.22 + … + N.2n = (N – 1) 2n+1 + 2 - Mathematics

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Question

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

Solution

Let the given statement be P(n), i.e.,

P(n): 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

For n = 1, we have

P(1): 1.2 = 2 = (1 – 1) 21+1 + 2 = 0 + 2 = 2, which is true.

Let P(k) be true for some positive integer k, i.e.,

1.2 + 2.22 + 3.22 + … + k.2k = (k – 1) 2k + 1 + 2 … (i)

We shall now prove that P(k + 1) is true.

Consider

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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APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 11 (2018 (Latest))
Chapter 4: Principle of Mathematical Induction
Q: 8 | Page no. 94
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Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: 1.2 + 2.22 + 3.22 + … + N.2n = (N – 1) 2n+1 + 2 Concept: Principle of Mathematical Induction.
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