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:
es002-1.jpg
Use summation notation to model and evaluate the sales for the first ten years. Explain your steps.
this is an arithmetic series with
a = 500
d = 10
look at
https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
To model and evaluate the sales for the first ten years using summation notation, we can start by identifying the given information and breaking it down step-by-step.
Given:
- The sales in the first month are $500.
- Sales increase at a rate of $10 each month.
To begin, let's define the variables:
- Sales in month one: S₁ = $500.
- Rate of increase per month: ΔS = $10.
Now, let's use summation notation to model the sales for the first ten years.
We want to model the sales for the first ten years, which is a total of 10 years x 12 months = 120 months. Therefore, our summation will have 120 terms.
The model using summation notation is as follows:
Sum of Sales (from month 1 to month 120) = Σ(S + iΔS)
Here, Σ represents the summation symbol, S is the initial sales, i is the index variable representing the month number (i = 0, 1, 2, ..., 119), and ΔS is the rate of increase per month.
Substituting the given values, our equation becomes:
Sum of Sales (from month 1 to month 120) = Σ($500 + i*$10) (where i = 0, 1, 2, ..., 119)
To evaluate the sales for the first ten years, we need to find the sum of the sales from month 1 to month 120.
Therefore, we can evaluate the sum by substituting 120 for the upper limit of the summation and performing the calculation:
Sum of Sales (from month 1 to month 120) = Σ($500 + i*$10) (where i = 0, 1, 2, ..., 119)
= $500 + 0*$10 + $500 + 1*$10 + $500 + 2*$10 + ... + $500 + 119*$10
Calculating each term of the summation and adding them up will give us the total sales for the first ten years, which can be computed as:
Sum of Sales (from month 1 to month 120) = $500 + $500 + ... + $500 + $500 + $10 + $20 + ... + $1,190
To simplify the calculation, we can use the formula for the sum of an arithmetic series:
Sum of Sales (from month 1 to month 120) = 120/2 * ($500 + $1,190)
= 60 * ($500 + $1,190) = $79,400
Therefore, the sales for the first ten years using the given model and summation notation is $79,400.
To use summation notation to model and evaluate the sales for the first ten years, first let's break down the given information:
Sales in the first month = $500
Sales increase at a rate of $10 each month
In order to find the total sales for the first ten years, we need to calculate the sum of the monthly sales.
Step 1: Determine the number of terms in the summation.
Since each year has 12 months, and we want to calculate the sales for the first ten years, we have a total of 10 * 12 = 120 terms.
Step 2: Write the expression for the terms in the summation.
The first month's sales is given as $500, and the sales increase by $10 each month. Therefore, the expression for the sales in the n-th month can be written as 500 + 10(n-1), where n is the month number.
Step 3: Write the summation notation.
Based on the above expression for the sales in the n-th month, the summation notation can be written as:
120
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/__
n=1
Step 4: Evaluate the summation.
To evaluate the summation, we substitute the values of n from 1 to 120 into the expression for the sales. We then find the sum of all these terms.
Using the formula for the sum of an arithmetic series:
S = n/2 * (first term + last term)
Substituting the values:
S = 120/2 * (500 + 500 + 10(120-1))
= 60 * (1000 + 10(119))
= 60 * (1000 + 1190)
= 60 * 2190
= 131400
Therefore, the total sales for the first ten years can be modeled and evaluated as $131,400.