Question

Use induction to prove that for any integer n, 1+5+9+…+(4n-3) = n(2n-1). What is being proven in the base case?

a) The sum of consecutive odd numbers is always an even number
b) The sum of consecutive odd numbers is always a μltiple of 3
c) The forμla holds true when n=1
d) The forμla holds true when n is a prime number

Answer 1

The base case in a proof by induction for the series 1+5+9+...+(4n-3) confirms that the formula n(2n-1) holds true when n=1. This is essential in establishing the validity of the formula for all subsequent integer values through the inductive step.

Step-by-step explanation:

The question involves mathematical induction to prove that the series 1+5+9+...+(4n-3) is equal to n(2n-1). In the base case of a proof by induction, we verify whether the given formula holds true for the first value of the integer sequence, which is typically n=1. Therefore, the correct answer to what is being proven in the base case is c) The formula holds true when n=1.

To test the base case, we substitute n=1 into the series and the formula:

• Series: 1
• Formula: 1(2*1-1) = 1

Since both the series and the formula yield the same value, the base case is confirmed, and the formula holds true for n=1. This is the starting point for the inductive proof, which then assumes the formula is true for some integer k, and proves it must also be true for k+1, and so on.