6. During his first week as a salesman, Barney sold 33 products. He challenged himself to sell 3 more products each successive week than he sold the week before. If Barney meets his goal with exactly 3 more products each week, how many products does he sell in week 15 ?

To solve the problem, we need to determine how many products Barney sells in week 15, given that he starts with 33 products in the first week and increases his sales by 3 products each week. 1. **Identify the known conditions:** - Products sold in week 1: \( P_1 = 33 \) - Increase in products sold each week: \( d = 3 \) - We need to find the number of products sold in week 15: \( P_{15} \) 2. **Establish the formula for the number of products sold in week \( n \):** The number of products sold in week \( n \) can be expressed as: \[ P_n = P_1 + (n - 1) \cdot d \] where \( P_1 \) is the number of products sold in the first week, \( d \) is the increase in sales per week, and \( n \) is the week number. 3. **Substituting the known values into the formula for week 15:** \[ P_{15} = 33 + (15 - 1) \cdot 3 \] 4. **Calculate the expression step by step:** - First, calculate \( 15 - 1 \): \[ 15 - 1 = 14 \] - Next, multiply by the increase \( d \): \[ 14 \cdot 3 = 42 \] - Finally, add this to the initial sales: \[ P_{15} = 33 + 42 = 75 \] Thus, the number of products Barney sells in week 15 is \( \boxed{75} \).