Given that P(n) is the equation XXXXXXXXXX+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we will prove that P (n) is true for all n ≥ 1 by induction. (a) Base case: i. Write P(1). ii. Show...

Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we will prove that P (n) is true for all n ≥ 1 by induction.

(a) Base case:
i. Write P(1).

ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side.

(b) Inductive hypothesis: Let k ≥ 1 be a natural number. Assume that P (k) is true. Write P (k).

(c) Inductive step:
i. WriteP(k+1).

ii. Use the assumption that P (k) is true to prove that P (k + 1) is true. Justify all of your steps.

(d) Explain why this shows that P (n) is true for all n ≥ 1.