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.

Show that the first of these conditions is satisfied by evaluating the left and right sides of the given statement for the first natural number.

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

\[ \square = \square \] (Simplify your answers.)

Solution

To use the principle of mathematical induction, the statement needs to satisfy two conditions: 1. The statement is true for the natural number 1. 2. If the statement is true for some natural number \( k \), it is also true for the next natural number \( k + 1 \). Now, let’s verify the first condition by evaluating the statement for \( n = 1 \). The given statement is: \[ 14 + 28 + 42 + \ldots + 14n = 7n(n+1) \] For \( n = 1 \), the left side is: \[ 14 \times 1 = 14 \] For the right side: \[ 7 \times 1 \times (1 + 1) = 7 \times 1 \times 2 = 14 \] So, both sides are equal, and the statement is true for \( n = 1 \). Therefore, the first condition is satisfied.