1 + 3 + 5 + 7 + …….. + (2n – 1) = n2, for all n ∈ N

1.	1 + 3 + 5 + 7 + …….. + (2n – 1) = n2, for all n ∈ N
Answer» Let P(n) : 1 + 3 + 5 + 7+……….. + (2n - 1) = n² For n = 1, LHS = 1, RHS = 12 = 1 ∴ LHS = RHS ∴ F(1) is true. Let upwards assume P(k) is true for some k ∈ N i.e., 1 + 3 + 5 + …. + (2K - 1) = kz Adding (k + 1)^th term = 2K + 1, on both sides, we get, 1 + 3 + 5 +……. +(2k -1)(2k + l) = k² + 2k + 1 = (k +1)² which is P(k +1) Thus, P(k) ⇒ P(k +1). Hence, by mathematical induction P(n) is true for all n∈N

Answer»

Let P(n) : 1 + 3 + 5 + 7+……….. + (2n - 1) = n²

For n = 1, LHS = 1, RHS = 12 = 1

∴ LHS = RHS

∴ F(1) is true.

Let upwards assume P(k) is true for some k ∈ N

i.e., 1 + 3 + 5 + …. + (2K - 1) = kz

Adding (k + 1)^th term = 2K + 1, on both sides, we get, 1 + 3 + 5 +……. +(2k -1)(2k + l) = k² + 2k + 1 = (k +1)²

which is P(k +1)

Thus, P(k) ⇒ P(k +1).

Hence, by mathematical induction P(n) is true for all n∈N

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