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प्रश्न
Answer the following:
Prove, by method of induction, for all n ∈ N
2 + 3.2 + 4.22 + ... + (n + 1)2n–1 = n.2n
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उत्तर
Let P(n) ≡ 2 + 3.2 + 4.22 + ... + (n + 1)2n–1 = n.2n-1 = n.2n, for all n ∈ N
Step I:
Put n = 1
L.H.S. = 2
R.H.S. = 1(21) = 2 = L.H.S.
∴ P(n) is true for n = 1
Step II:
Let us consider that P(n) is true for n = k
∴ 2 + 3.2 + 4.22 + … + (k + 1)2k–1 = k.2k …(i)
Step III:
We have to prove that P(n) is true for n = k + 1
i.e., to prove that
2 + 3.2 + 4.22 + …. + (k + 2)2k = (k + 1)2k+1
L.H.S. = 2 + 3.2 + 4.22 + …. + (k + 2)2k
= 2 + 3.2 + 4.22 + …. + (k + 1)2k–1 + (k + 2)2k
= k.2k + (k + 2).2k …[From (i)]
= (k + k + 2).2k
= (2k + 2). 2k
= (k + 1).2.2k
= (k + 1). 2k+1
= R.H.S.
∴ P(n) is true for n = k + 1
Step IV:
From all steps above by the principle of mathematical induction, P(n) is true for all n ∈ N.
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