The total salary earned over twenty years is
[tex] \sum_{k=1}^{20}[50,000+(k-1) 1,500][/tex]
Answer (2)
Break down the summation into simpler parts: ∑ k = 1 20 50 , 000 + 1 , 500 ∑ k = 1 20 k − 1 , 500 ∑ k = 1 20 1 .
Calculate the individual summations: ∑ k = 1 20 50 , 000 = 1 , 000 , 000 , ∑ k = 1 20 k = 210 , and ∑ k = 1 20 1 = 20 .
Substitute the values back into the expression: 1 , 000 , 000 + 1 , 500 ( 210 ) − 1 , 500 ( 20 ) .
Simplify to find the total salary: 1 , 285 , 000 .
Explanation
Understanding the Problem We are given the expression for the total salary earned over twenty years as a summation: k = 1 ∑ 20 [ 50 , 000 + ( k − 1 ) 1 , 500 ] Our goal is to calculate the value of this summation.
Breaking Down the Summation First, we can split the summation into separate parts: k = 1 ∑ 20 50 , 000 + k = 1 ∑ 20 ( k − 1 ) 1 , 500 = k = 1 ∑ 20 50 , 000 + 1 , 500 k = 1 ∑ 20 ( k − 1 ) We can further simplify the second summation by separating the terms: 1 , 500 k = 1 ∑ 20 ( k − 1 ) = 1 , 500 ( k = 1 ∑ 20 k − k = 1 ∑ 20 1 ) So the expression becomes: k = 1 ∑ 20 50 , 000 + 1 , 500 k = 1 ∑ 20 k − 1 , 500 k = 1 ∑ 20 1
Calculating Individual Summations Now, let's evaluate each summation separately. The first summation is simply: k = 1 ∑ 20 50 , 000 = 20 × 50 , 000 = 1 , 000 , 000 The second summation involves the sum of the first 20 integers. We can use the formula for the sum of the first n integers, which is 2 n ( n + 1 ) . In this case, n = 20 , so: k = 1 ∑ 20 k = 2 20 ( 20 + 1 ) = 2 20 × 21 = 210 The third summation is: k = 1 ∑ 20 1 = 20
Calculating the Total Salary Now, substitute these values back into the expression: k = 1 ∑ 20 50 , 000 + 1 , 500 k = 1 ∑ 20 k − 1 , 500 k = 1 ∑ 20 1 = 1 , 000 , 000 + 1 , 500 ( 210 ) − 1 , 500 ( 20 ) = 1 , 000 , 000 + 315 , 000 − 30 , 000 = 1 , 285 , 000 Therefore, the total salary earned over twenty years is $1,285,000.
Final Answer The total salary earned over twenty years is $1,285,000.
Examples
Understanding how to calculate the total salary earned over a period of time is crucial for financial planning. For instance, if you are planning for retirement, you need to estimate your total earnings to determine how much you can save and invest. This calculation helps in projecting your future income and making informed decisions about investments, expenses, and savings. It's also useful in evaluating different job offers by comparing the total compensation over the expected duration of employment. For example, if you earn $50,000 in the first year and your salary increases by $1,500 each year for 20 years, the total earnings can be calculated as shown above, resulting in a total of $1,285,000.
The total salary earned over twenty years is calculated to be $1,285,000. This is achieved by breaking down the salary into fixed and variable components and summing them over the given period. Understanding such arithmetic sequences is crucial in financial calculations and planning.
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