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Prove by the method of induction, for all n ∈ N.

5 + 52 + 53 + ….. + 5n\(\frac54\)(5n - 1)

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Let P(n) ≡ 5 + 52 + 53 +…..+ 5n = \(\frac54\)(5n – 1), for all n ∈ N.

Step I:

Put n = 1

L.H.S. = 5

R.H.S. = \(\frac54\)(51 – 1) = 5

∴ L.H.S. = R.H.S.

∴ P(n) is true for n = 1.

Step II:

Let us assume that P(n) is true for n = k.

∴ 5 + 52 + 53 + ….. + 5k = \(\frac54\)(5k – 1) …….(i)

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

∴ P(n) is true for n = k + 1.

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

∴ 5 + 52 + 53 + … + 5n = \(\frac54\)(5 – 1), for all n ∈ N.

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