Its part e I am stuck on

Q: Let P(n) be the statement that 1^3 + 2^3 +….n^3 = (n(n + 1) / 2)² for the positive

integer n.

a.) What is the statement P(1)?

b.) Show that P(1), 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?

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

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A:

a.) P(n) = (n(n + 1) / 2)²

P(1) = (1(1 + 1) / 2)²

= (2 / 2)²

= 1

b.) Basis Step:

we will let n = 1

n^3 = (n(n + 1) / 2)²

1^3 = (1(1 + 1) / 2)²

1 = (2 / 2)²

1 = 1

Since both are equal, the basis step hold.

c.) Inductive Hypothesis:

This is the statement 1^3 + 2^3 +….k^3 = (k(k + 1) / 2)²

d.) I have to prove that k>1 and P(k) implies P(k + 1).

Or.

1^3 + 2^3 +….k^3 + (k + 1)^3 = ((k + 1)((k + 1) + 1) / 2)²

1^3 + 2^3 +….k^3 + (k + 1)^3 = (((k + 1)((k + 2)) / 2)²

e.)

1^3 + 2^3 +….k^3 + (k + 1)^3 = ((k + 1)((k + 1) + 1) / 2)² + (k + 1)

1^3 + 2^3 +….k^3 + (k + 1)^3 = ((k + 1)² · ((k + 2)²) / 4 + (k + 1)

1^3 + 2^3 +….k^3 + (k + 1)^3 = ((k + 1)² · (k + 2))² + 4(k + 1)