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Chapter 11: Problem 29

Evaluate each series. $$\sum_{i=1}^{5}(2 i+1)$$

Short Answer

Expert verified
The value of the series \[\begin{equation}\sum_{i=1}^{5}(2i+1)\end{equation}\] is 35.

Step by step solution

01

Understand the Series

The given series is \[\begin{equation}\sum_{i=1}^{5}(2i+1)\end{equation}\]This means we need to evaluate the sum of the expression \[\begin{equation}2i+1\end{equation}\]for each value of i from 1 to 5.
02

Substitute Values of i

Substitute the values of i from 1 to 5 into the expression \[\begin{equation}2i+1\end{equation}\]We get:i=1: \[\begin{equation}2(1) + 1 = 3\end{equation}\]i=2: \[\begin{equation}2(2) + 1 = 5\end{equation}\]i=3: \[\begin{equation}2(3) + 1 = 7\end{equation}\]i=4: \[\begin{equation}2(4) + 1 = 9\end{equation}\]i=5: \[\begin{equation}2(5) + 1 = 11\end{equation}\]
03

Add the Terms

Now add the individual results from the previous step:\[\begin{equation}3 + 5 + 7 + 9 + 11\end{equation}\]
04

Calculate the Result

Calculate the sum:\[\begin{equation}3 + 5 = 8\end{equation}\]\[\begin{equation}8 + 7 = 15\end{equation}\]\[\begin{equation}15 + 9 = 24\end{equation}\]\[\begin{equation}24 + 11 = 35\end{equation}\]

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Key Concepts

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

arithmetic series
Let's begin by understanding what an arithmetic series is. An arithmetic series is the sum of the terms of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is always constant. This difference is known as the common difference.

An example of an arithmetic sequence would be: 2, 4, 6, 8, 10... Here, the common difference is 2.

When we sum the numbers in this sequence, we form an arithmetic series. In our specific problem, each term was created by the formula 2i + 1, which results in an arithmetic sequence when calculated for consecutive values of i. The series we formed was 3, 5, 7, 9, 11, which also has a common difference of 2.
sigma notation
Sigma notation is a concise and efficient way to represent the sum of a series. It uses the Greek letter \(\sum\)\(\sum\) to indicate summation. The general format is:
\[ \sum_{i=a}^{b} \text{{expression involving }} i \]
Where:
  • a is the starting value of the index i
  • b is the ending value of the index i
  • expression involving i is the formula that generates the terms to be summed.
In our problem, the sigma notation \[ \sum_{i=1}^{5}(2i+1)\]\(\sum_{i=1}^{5}(2i+1)\) tells us to evaluate the expression 2i + 1 for each value of i from 1 to 5 and then sum these evaluated terms together. Sigma notation simplifies the representation of such summations and makes it clear at a glance how many terms there are and their range.
step-by-step evaluation
Step-by-step evaluation is a methodical approach to solving a problem. Let's recap the steps involved in solving our arithmetic series problem using this approach:

First, we identified our series in sigma notation: \(\sum_{i=1}^{5}(2i+1)\)
Next, we expanded the series by substituting the values of i from 1 to 5:
  • \(2(1) + 1 = 3\)
  • \(2(2) + 1 = 5\)
  • \(2(3) + 1 = 7\)
  • \(2(4) + 1 = 9\)
  • \(2(5) + 1 = 11\)

Then, we summed up these values:
\(3 + 5 + 7 + 9 + 11\)
Finally, we performed the calculations:
  • \(3 + 5 = 8\)
  • \(8 + 7 = 15\)
  • \(15 + 9 = 24\)
  • \(24 + 11 = 35\)
The total sum is 35. By following this clear sequence of steps, we simplified the problem and ensured accuracy at each stage.

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