The question asks which of the following statements is a valid conclusion one might reach in a proof by induction. The options are: a. The sum of odd numbers is $2n$. b. The statement holds only for prime numbers. c. The sum of squares formula does not hold. d. The sum of the first $n$ integers is $n(n+1)/2$. e. The statement is only true for even numbers.

1. Problem Description

The question asks which of the following statements is a valid conclusion one might reach in a proof by induction. The options are:

a. The sum of odd numbers is $2n$.

b. The statement holds only for prime numbers.

c. The sum of squares formula does not hold.

d. The sum of the first $n$ integers is $n(n+1)/2$.

e. The statement is only true for even numbers.

2. Solution Steps

We need to consider what types of conclusions can be reached through proof by induction. Proof by induction is typically used to prove that a statement is true for all natural numbers (or for all integers greater than or equal to some base case). The conclusions will usually be in the form of a formula or a property that holds for all $n$.

a. The sum of the first $n$ odd numbers is $1 + 3 + 5 + \dots + (2n-1)$. We can prove by induction that this sum is equal to $n^2$. The statement "The sum of odd numbers is $2n$" is incorrect, as the sum of odd numbers depends on how many odd numbers are being summed.

b. Proof by induction often aims to show a statement holds for all integers greater or equal to some base. Therefore, stating that a statement only holds for prime numbers is not a typical conclusion of proof by induction.

c. Similar to b, this option discusses a formula not holding. Proof by induction attempts to prove the formula does hold. Therefore, this is not a common type of conclusion from proof by induction.

d. The sum of the first $n$ integers is $1 + 2 + 3 + \dots + n$. This is a common example used for induction. The formula for this sum is $n(n+1)/2$. So this is a valid formula one might obtain from proof by induction.

e. Similar to b, proof by induction seeks to prove a statement to be true for all integers greater or equal to a base case. Therefore stating that the statement is only true for even numbers is not a valid conclusion from proof by induction.

Therefore, the correct answer is d.

3. Final Answer

d. The sum of the first $n$ integers is $n(n+1)/2$