Question: Simplify each side of the statement for the appropriate value of \( n \). \[ 7 + 15 + 23 +…

Simplify each side of the statement for the appropriate value of \( n \).

\[ 7 + 15 + 23 + \ldots + (8n - 1) = n(4n + 3) \]

\[ \boxed{\phantom{0}} = \boxed{\phantom{0}} \]

Solution

To solve the problem, we need to simplify each side of the equation given and find the appropriate value of \( n \). The expression on the left side is an arithmetic sequence: \[ 7 + 15 + 23 + \ldots + (8n - 1) \] First, identify the first term \( a \) and the common difference \( d \). The first term \( a = 7 \). The common difference \( d = 15 - 7 = 8 \). The nth term of an arithmetic sequence can be given by: \[ a_n = a + (n-1) \cdot d \] Set the nth term equal to \( 8n - 1 \): \[ a + (n-1) \cdot d = 8n - 1 \] \[ 7 + (n-1) \cdot 8 = 8n - 1 \] Simplify to check consistency: \[ 7 + 8n - 8 = 8n - 1 \] \[ 8n - 1 = 8n - 1 \] The nth term formula is consistent. Now, find the number of terms \( n \) when the nth term equals \( 8n - 1 \). For the sum of the sequence: The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is: \[ S_n = \frac{n}{2} \cdot (a + l) \] where \( l = 8n - 1 \) is the last term. \[ S_n = \frac{n}{2} \cdot (7 + (8n - 1)) \] \[ S_n = \frac{n}{2} \cdot (8n + 6) \] Simplify: \[ S_n = n(4n + 3) \] Compare this with the right side of the equation: \[ n(4n + 3) \] The simplified sums on both sides of the equation yield the same expression, confirming the consistency. Therefore, the appropriate value of \( n \) that makes both sides equal is any positive integer, as the expressions are always equivalent. However, substitute \( n = 5 \) since finding the simplest specific value typically involves trial and it satisfies: \[ 8 \cdot 5 - 1 = 39 \] Finally, the appropriate value of \( n \) for computation: \[ n = 5 \]