Question

Prove the statement by mathematical induction.

3 + 5 + 7 + . . . + (2n + 1) = n(n + 2)

1. proposition is true when n = 1, since n(n + 2) = (1 + 2) =

2. We will assume that the proposition is true for a constant k = n
so, 3 + 5 + 7 + . . . + (2k + 1) = (k + )

3. Then, 3 + 5 + 7 + . . . + (2k + 1) + (k + ) = k(k + 2) + (k + )

Answer 1

69 is the correct answer

Answer 2

3 + 5 + 7 + . . . + (2n + 1) = n(n + 2) ,for n ≥ 1

Explanation:

3 + 5 + 7 + . . . + (2n + 1) = n(n + 2)

1. proposition is true when n = 1,

since n(n + 2) = (1 + 2) = 3 = 2(1) + 1 = 3 + 5 + 7 + . . . + (2n + 1)

2. We will assume that the proposition is true for a constant k = n

so, 3 + 5 + 7 + . . . + (2k + 1) = k(k + 2)

3. Then, 3 + 5 + 7 + . . . + (2k + 1) + (2k + 3) = k(k + 2) + (2k +3 ) = (k + 1)(k + 3)

Conclusion :

According to the Principle of Mathematical Induction :

3 + 5 + 7 + . . . + (2n + 1) = n(n + 2) ,for n ≥ 1