Question

What will be P(k + 1) for P(n) = n^3 (n + 1)?

(a) (k + 1)^4

(b) k^4 + 5k^3 + 9k^2 + 7k + 2

(c) k^4 + 6k^3 + 9k^2 + 7k + 2

(d) k^4 + 3k^3 + 9k^2 + 6k + 2

I have been asked this question in my homework.

I would like to ask this question from The Principle of Mathematical Induction in section Principle of Mathematical Induction of Mathematics – Class 11

Answer 1

Correct choice is (b) k^4 + 5k^3 + 9k^2 + 7k + 2

The best I can explain: P(n) = n^3 (n + 1)

P(k + 1) = (k + 1)^3 (k + 1 + 1)

P(k + 1) = (k^3 + 3k^2 + 3k + 1) (k + 2)

P(k + 1) = k^4 + 3k^3 + 3k^2 + k + 2k^3 + 6k^2 + 6k + 2

P(k + 1) = k^4 + 5k^3 + 9k^2 + 7k + 2