A store has sales of $500 in their first month. If sales increase at a rate of $10 each month, they can be modeled by this equation:
[tex]a_n=500+(k-1) 10[/tex]
Use summation notation to model and evaluate the sales for the first ten years. Explain your steps.
Answer (2)
Model the total sales for the first 120 months using summation notation: ∑ k = 1 120 ( 500 + ( k − 1 ) 10 ) .
Expand the summation: ∑ k = 1 120 ( 490 + 10 k ) = 490 ∑ k = 1 120 1 + 10 ∑ k = 1 120 k .
Evaluate the summations: ∑ k = 1 120 1 = 120 and ∑ k = 1 120 k = 7260 .
Calculate the total sales: 490 ( 120 ) + 10 ( 7260 ) = 131400 . The total sales for the first ten years are 131400 .
Explanation
Understanding the Problem We are given that the sales in the first month are $500, and the sales increase by $10 each month. The sales in month k can be modeled by the equation a k = 500 + ( k − 1 ) 10 . We want to find the total sales for the first ten years, which is equivalent to 120 months. We will use summation notation to model and evaluate the total sales.
Setting up the Summation The total sales for the first 120 months can be represented by the summation: k = 1 ∑ 120 a k = k = 1 ∑ 120 ( 500 + ( k − 1 ) 10 ) We need to evaluate this summation.
Expanding the Summation First, let's expand the expression inside the summation: k = 1 ∑ 120 ( 500 + 10 k − 10 ) = k = 1 ∑ 120 ( 490 + 10 k ) Now, we can separate the summation into two parts: k = 1 ∑ 120 490 + k = 1 ∑ 120 10 k = 490 k = 1 ∑ 120 1 + 10 k = 1 ∑ 120 k
Evaluating the Summations We know that ∑ k = 1 120 1 = 120 and ∑ k = 1 120 k = 2 120 ( 120 + 1 ) = 2 120 ( 121 ) = 60 ( 121 ) = 7260 .
Substituting these values back into the expression, we get: 490 ( 120 ) + 10 ( 7260 ) = 58800 + 72600 = 131400
Final Answer Therefore, the total sales for the first ten years are $131,400.
Examples
Summation notation is a powerful tool used in various fields like finance and economics to model and calculate cumulative values over a period. For instance, if you're investing a fixed amount every month and earning a certain interest rate, summation notation can help you calculate the total value of your investment after a specific number of years. Similarly, businesses use it to forecast total revenue or costs over multiple periods, aiding in budgeting and strategic planning. Understanding and applying summation notation provides a clear and concise way to analyze and predict outcomes in scenarios involving repeated additions.
The total sales for the first ten years, calculated using summation notation, amount to $131,400. This was found by modeling the monthly sales and evaluating the sum for 120 months. The step-by-step evaluation involved using mathematical summation formulas for constant and linear sequences.
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