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.