What is the sum of the series?
[tex]$\sum_{k=1}^{84}(3 k-8)$[/tex]
Answer (2)
Split the summation: ∑ k = 1 84 ( 3 k − 8 ) = ∑ k = 1 84 3 k − ∑ k = 1 84 8 .
Use the formula for the sum of the first n integers: ∑ k = 1 84 k = 2 84 ( 85 ) .
Calculate the sum: 3 × 2 84 × 85 − 8 × 84 = 10710 − 672 = 10038 .
The sum of the series is: 10038 .
Explanation
Understanding the Problem We are asked to find the sum of the series ∑ k = 1 84 ( 3 k − 8 ) . This means we need to add up the values of the expression 3 k − 8 for each integer k from 1 to 84.
Splitting the Summation We can split the summation into two parts using the properties of summation: k = 1 ∑ 84 ( 3 k − 8 ) = k = 1 ∑ 84 3 k − k = 1 ∑ 84 8
Factoring Constants We can factor out the constant 3 from the first summation: k = 1 ∑ 84 3 k = 3 k = 1 ∑ 84 k The second summation is simply adding 8 a total of 84 times: k = 1 ∑ 84 8 = 8 × 84
Using the Sum of Integers Formula We can use the formula for the sum of the first n integers, which is given by: k = 1 ∑ n k = 2 n ( n + 1 ) In our case, n = 84 , so we have: k = 1 ∑ 84 k = 2 84 ( 84 + 1 ) = 2 84 × 85
Combining the Results Now we substitute the results back into our original expression: k = 1 ∑ 84 ( 3 k − 8 ) = 3 k = 1 ∑ 84 k − k = 1 ∑ 84 8 = 3 × 2 84 × 85 − 8 × 84
Final Calculation Now we perform the calculation: k = 1 ∑ 84 ( 3 k − 8 ) = 3 × 2 84 × 85 − 8 × 84 = 3 × ( 42 × 85 ) − 672 = 3 × 3570 − 672 = 10710 − 672 = 10038
Conclusion Therefore, the sum of the series is 10038.
Examples
Imagine you are building a staircase where the height of each step increases linearly. The first step is 5 inches high, the second is 8 inches, the third is 11 inches, and so on. You want to know the total height you'll climb after 84 steps. This problem is similar to summing an arithmetic series, where each term represents the height of a step. By calculating the sum, you can determine the total vertical distance covered by the staircase, which is crucial for planning and construction.
To find the sum ∑ k = 1 84 ( 3 k − 8 ) , we split the summation, calculate ∑ k = 1 84 k , and then evaluate the total. The final result is 10038 .
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