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Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: 1.3 + 2.3^3 + 3.3^3 +...+ N.3^N = ((2n -1)3^(N+1) + 3)/4

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प्रश्न

Prove the following by using the principle of mathematical induction for all n ∈ N

1.3 + 2.3^3 + 3.3^3  +...+ n.3^n = `((2n -1)3^(n+1) + 3)/4`
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उत्तर

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

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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