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Chapter 14: Problem 12

Evaluate the sums. \(\sum_{i=1}^{4} 2\)

Short Answer

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Question: Evaluate the sum of the expression \(\sum_{i=1}^{4} 2\). Answer: The sum of the expression \(\sum_{i=1}^{4} 2\) is 8.

Step by step solution

01

Understand the sum notation

The given expression is \(\sum_{i=1}^{4} 2\). Here, the summation symbol \(\sum\) is telling us to add the terms of a series. The limits of summation are \(i=1\) (starting value) and \(4\) (ending value). The expression within the summation, i.e., the term being added, is the constant \(2\).
02

Write the sum explicitly

Since the term being added is constant, we can write out all terms of the series being summed explicitly. The sum becomes: $$2 + 2 + 2 + 2$$ Notice that we have four terms (\(4\) being the ending value of the limit) of the series, and each term is the constant \(2\).
03

Evaluate the sum

Adding all the terms of the constant series, we get: $$2 + 2 + 2 + 2 = 8$$ Now, we have found the value of the given sums for the expression. The result is \(\sum_{i=1}^{4} 2 = 8\).

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Key Concepts

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

Summation Notation
When you encounter \( \sum_{i=1}^{4} 2 \), this is an example of summation notation, a concise way to express sums of sequences. This notation involves the sigma symbol \( \sum \), which instructs us to sum up certain values.
The expression specifies a range and a term to add. The values that \( i \) takes on are specified by the limits of summation: the lower limit, \( i=1 \), and the upper limit, \( i=4 \). Essentially, this tells us to sum the value \( 2 \), starting from \( i=1 \) and ending at \( i=4 \).
  • The number under the \( \sum \) (\( i=1 \)) is where you start counting.
  • The number above the \( \sum \) (\( 4 \)) is where you stop.
  • The expression following \( \sum \) (\( 2 \)) is what you're summing.
This efficient notation simplifies working with sequences and helps in quickly setting up equations for evaluation.
Series Evaluation
Evaluating a series is straightforward when all you have is a constant added multiple times. In our problem, each term in the series is the constant \( 2 \). For each value of \( i \) from \( 1 \) to \( 4 \), the term \( 2 \) is repeated four times.
This can be written explicitly as:
\[ 2 + 2 + 2 + 2 \]
In this simple scenario, our focus is on adding up these terms. No complex calculations, just straightforward addition.
Through explicit writing, you gain clarity on how many terms are present and what each term consists of, ensuring a thorough understanding of this sum.
Constant Addition
The beauty of constant addition is its simplicity. If you're adding the same number repeatedly, it becomes a basic multiplication problem.
Here's how: Since each of the four terms is \( 2 \), you perform the addition:
\[ 2 + 2 + 2 + 2 = 8 \]
Alternatively, recognize that you're adding \( 2 \) four times, which simplifies to multiplying:
\( 4 \times 2 = 8 \).
This approach highlights how constant repetition in sums translates to multiplication.
  • Convert addition of repeated terms into multiplication.
  • Multiply the constant by the number of terms.
This technique simplifies evaluating sums, especially as sequences grow larger, making higher calculations more manageable.

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Most popular questions from this chapter

Write out the sums. (You do not need to evaluate them.) \(\sum_{j=1}^{4}(-1)^{j}\)

Are the expressions equivalent? \(\sum_{i=1}^{5} a_{i}\) and \(a_{5}+a_{4}+a_{3}+a_{2}+a_{1}\)

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