Question: Use the Principle of Mathematical Induction to show that the following statement is true for all…

Use the Principle of Mathematical Induction to show that the following statement is true for all natural numbers \( n \).

\[ 14 + 28 + 42 + \ldots + 14n = 7n(n + 1) \]

What two conditions must the given statement satisfy to prove that it is true for all natural numbers? Select all that apply.

• The statement is true for the natural number 1.
• If the statement is true for some natural number \( k \), it is also true for the next natural number \( k + 1 \).
• The statement is true for any two natural numbers \( k \) and \( k + 1 \).
• If the statement is true for the natural number 1, it is also true for the next natural number 2.

Solution

To use the Principle of Mathematical Induction to prove that a statement is true for all natural numbers \( n \), the statement must satisfy the following two conditions: 1. Base Case: The statement is true for the natural number \( n = 1 \). 2. Inductive Step: If the statement is true for some natural number \( k \), it is also true for the next natural number \( k + 1 \). Thus, the correct boxes to check are: - The statement is true for the natural number 1. - If the statement is true for some natural number \( k \), it is also true for the next natural number \( k + 1 \).