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Chapter 14: Problem 67

Find each sum. $$ \sum_{k=4}^{4}(2 k+4) $$

Short Answer

Expert verified
The sum is 12.

Step by step solution

01

Identify the Limits of the Sum

Here, the index of summation, denoted as \( k \), starts at 4 and ends at 4. This means there is only one term to be calculated in the sum.
02

Plug the Value into the Expression

Substitute \( k = 4 \) into the expression \( 2k + 4 \). This gives us:\[2(4) + 4\]
03

Calculate the Expression

Simplify the expression by performing the multiplication and addition:\[2(4) + 4 = 8 + 4 = 12\]
04

Write the Final Answer

Since the sum has only one term, the value of the sum is simply the value of the expression we just calculated, which is 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits of Summation
In summation notation, the limits of summation define the range over which the sum is calculated. They tell us the starting and ending values for the index of summation. In the exercise, the limits are from 4 to 4, meaning the index starts and ends at the same value. This indicates that only one term of the expression needs to be evaluated. Essentially, the limit of summation here doesn't involve a series of numbers, just a single evaluation at one point, precisely when the index, represented by \( k \), is 4.
Index of Summation
The index of summation is the variable, commonly denoted by symbols like \( i \), \( j \), or \( k \), which changes values within the specified limits of summation. It's essential to identify this index correctly as it dictates which values you plug into the given expression. In the given exercise, the index is \( k \), and since our limits are defined as \( k = 4 \) to \( k = 4 \), it means we only consider \( k = 4 \) when calculating the sum. The index serves as a placeholder that runs through the terms of a sequence, guiding us in evaluating the sum step by step.
Evaluate Expression
Evaluating the expression involves substituting the value of the index of summation into the given formula. After identifying that \( k = 4 \) is the only term due to the limits of summation, we substitute it into the expression \( 2k + 4 \). By performing the arithmetic operations, we get \( 2(4) + 4 = 8 + 4 = 12 \).
  • First, substitute \( k = 4 \) into the expression.
  • Next, calculate \( 2(4) \) which equals 8.
  • Add 4 to 8 to obtain the final result, 12.
The final sum is thus 12, as there was only a single term to add.

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