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Question
Prove by method of induction, for all n ∈ N:
5 + 52 + 53 + .... + 5n = `5/4(5^"n" - 1)`
Solution
Let P(n) ≡ 5 + 52 + 53 + .... + 5n = `5/4(5^"n" - 1)`, for all n ∈ N
Step I:
Put n = 1
L.H.S. = 5
R.H.S. = `5/4(5^1 - 1)` = 5 = L.H.S.
∴ P(n) is true for n = 1.
Step II:
Let us consider that P(n) is true for n = k.
∴ 5 + 52 + 53 + …. + 5k = `5/4(5^"k" - 1)` ...(i)
Step III:
We have to prove that P(n) is true for n = k + 1 i.e., to prove that
5 + 52 + 53 + …. + 5k+1 = `5/4(5^("k"+1) - 1)`
L.H.S. = 5 + 52 + 53 + …. + 5k+1
= 5 + 52 + 53 + …. + 5k + 5k+1
= `5/4(5^"k" - 1) + 5^("k"+1)` ...[From (i)]
= `(5.5^"k" - 5 + 4.5^("k"+1))/4`
= `(5^("k"+1) + 4.5^("k"+1) - 5)/4`
= `(5.5^("k"+1) - 5)/4`
= `5/4(5^("k" + 1) - 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.
∴ 5 + 52 + 53 + .... + 5n = `5/4(5^"n" - 1)`, for all n ∈ N.
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