Question

Use mathematical induction to show that: 1³ + 3³ +5³+...........+(2n+1)³ = (n+1)³ (2n² +4n+1) whenever n is a positive integer.

b) Use mathematical induction to show:

[Please provide the specific statement or expression you'd like to be proven in part b], whenever
n is a positive integer.
You can proceed to solve both parts of the question.

Answer 1

To prove the given statement using mathematical induction, we first establish that the statement is true for the base case, which is when n = 1.

Step-by-step explanation:

To prove the given statement using mathematical induction, we first establish that the statement is true for the base case, which is when n = 1. Substituting n = 1 into the expression, we have 1³ = (1+1)³(2(1)² + 4(1) + 1), which simplifies to 1 = 8. This is indeed true, so the base case holds.

Next, we assume that the statement is true for an arbitrary positive integer k, denoted as P(k). That is, 1³ + 3³ + 5³ + ... + (2k+1)³ = (k+1)³(2k² + 4k + 1).

Now, we need to prove that the statement is true for the next positive integer, which is k+1. We add (2(k+1)+1)³ to both sides of the assumed statement P(k). After some algebraic manipulation, we can show that the expression simplifies to (k+1+1)³(2(k+1)² + 4(k+1) + 1). Therefore, by mathematical induction, the statement is true for all positive integers n.