Question

Let P (n) be the statement that 1³ + 2³ + · · · + n³ = (n(n + 1)/2)² for the positive integer n.

a) What is the statement P (1)?

b) Show that P (1) is true, completing the basis step of the proof. c) What is the inductive hypothesis?

d) What do you need to prove in the inductive step?

e) Complete the inductive step, identifying where you use the inductive hypothesis.

f ) Explain why these steps show that this formula is true whenever n is a positive integer.

Answer 1

a) The statement P(1) is:

1³ = (1(1 + 1)/2)²

b) To show that P(1) is true, we substitute the value of n = 1 into the equation:

1³ = (1(1 + 1)/2)²

The left-hand side of the equation is:

1³ = 1

The right-hand side of the equation is:

(1(1 + 1)/2)² = (1(2)/2)² = (2/2)² = 1² = 1

Therefore, the equation is true for n = 1.

c) The inductive hypothesis is that the statement P(k) is true for some positive integer k. In other words, assume that:

1³ + 2³ + · · · + k³ = (k(k + 1)/2)²

d) In the inductive step, we need to prove that if the statement P(k) is true, then the statement P(k + 1) is also true. In other words, we need to prove that:

1³ + 2³ + · · · + k³ + (k + 1)³ = ((k + 1)((k + 1) + 1)/2)²

e) Using the inductive hypothesis, assume that the statement P(k) is true:

1³ + 2³ + · · · + k³ = (k(k + 1)/2)²

Now, we need to prove the statement for P(k + 1):

1³ + 2³ + · · · + k³ + (k + 1)³ = ((k + 1)((k + 1) + 1)/2)²

Adding (k + 1)³ to both sides of the equation:

1³ + 2³ + · · · + k³ + (k + 1)³ = (k(k + 1)/2)² + (k + 1)³

Expanding and simplifying the right-hand side:

1³ + 2³ + · · · + k³ + (k + 1)³ = (k²(k + 1)²/4) + (k + 1)³

Combining the terms with a common denominator:

1³ + 2³ + · · · + k³ + (k + 1)³ = (k²(k + 1)² + 4(k + 1)³)/4

Simplifying further:

1³ + 2³ + · · · + k³ + (k + 1)³ = ((k + 1)(k² + 4(k + 1)))/4

Expanding and simplifying:

1³ + 2³ + · · · + k³ + (k + 1)³ = ((k + 1)(k² + 4k + 4))/4

Factoring out (k + 1)²:

1³ + 2³ + · · · + k³ + (k + 1)³ = ((k + 1)²(k + 4))/4

Simplifying further:

1³ + 2³ + · · · + k³ + (k + 1)³ = ((k + 1)((k + 1) + 1)/2)²

This is the same as the right-hand side of the statement P(k + 1), which completes the inductive step.

f) The basis step (P(1)) has been shown to be true, and the inductive step