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Magazine Chapter 14: Problem 70
Find each sum. $$ \sum_{k=2}^{5}(3 k-5) $$
Short Answer
Expert verified
The sum is 22.
Step by step solution
01
Understanding the sum notation
The notation \( \sum_{k=2}^{5} (3k - 5) \) represents the sum of the expression \( 3k - 5 \) for each integer \( k \) from 2 to 5, inclusive. We will evaluate the expression for each value of \( k \) and sum the results.
02
Calculate for k = 2
Substitute \( k = 2 \) into the expression \( 3k - 5 \): \[ 3(2) - 5 = 6 - 5 = 1 \]
03
Calculate for k = 3
Substitute \( k = 3 \) into the expression \( 3k - 5 \): \[ 3(3) - 5 = 9 - 5 = 4 \]
04
Calculate for k = 4
Substitute \( k = 4 \) into the expression \( 3k - 5 \): \[ 3(4) - 5 = 12 - 5 = 7 \]
05
Calculate for k = 5
Substitute \( k = 5 \) into the expression \( 3k - 5 \): \[ 3(5) - 5 = 15 - 5 = 10 \]
06
Add all results
Now, add all the results from each calculation: \[ 1 + 4 + 7 + 10 = 22 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Algebraic Expression
An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. It does not have an equality sign like an equation. In the exercise, the expression \(3k - 5\) is what we focus on when working through the steps.
Here are some key features of algebraic expressions:
Here are some key features of algebraic expressions:
- Variables: Symbols like \(k\) that represent unknown or variable quantities.
- Coefficients: Numbers that multiply the variables, such as 3 in \(3k\).
- Constants: Fixed values that do not change, like -5 in the expression.
- Operators: Symbols like +, -, *, and / that indicate the operations to perform.
Integer Sequence
An integer sequence is simply a list of numbers in a specific order. In mathematics, these sequences often follow a certain pattern. In the exercise, the integer sequence covers the values of \(k\) starting from 2 up to 5. Here鈥檚 how they were used:
Substitute each value of \(k\) in the expression \(3k - 5\) to get different results.
Substitute each value of \(k\) in the expression \(3k - 5\) to get different results.
- When \(k = 2\): \( 3(2) - 5 = 1 \)
- When \(k = 3\): \( 3(3) - 5 = 4 \)
- When \(k = 4\): \( 3(4) - 5 = 7 \)
- When \(k = 5\): \( 3(5) - 5 = 10 \)
Sigma Notation
Sigma notation, represented by the Greek letter \(\Sigma\), is a compact way to write the sum of a series of terms. It is especially handy for expressing long sums concisely. In our exercise, \( \sum_{k=2}^{5} (3k - 5) \) is a perfect example of using sigma notation.
Here is how it works:
Here is how it works:
- Summation Limits: The numbers below and above \(\Sigma\) (here, 2 and 5) are the limits, indicating the starting and ending values of the variable \(k\).
- Expression: The expression \(3k - 5\) is evaluated for each integer \(k\) from the lower limit to the upper limit.
- Increment: \(k\) increases by 1 (or by any other specified increment if mentioned) until it reaches the upper limit.
