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Chapter 10: Problem 38

Find the sum. $$\sum_{i=0}^{7}(2+4 i)$$

Short Answer

Expert verified
The sum of the series is 128.

Step by step solution

01

Identify the first and last term

The first term is when \(i=0\) and the last term in the series is when \(i=7\). So we replace \(i\) with those values in the expression \(2+4i\) to find the first term \(a\) and last term \(l\). For \(i=0\), \(a=2+4*0=2\). For \(i=7\), \(l=2+4*7=30\).
02

Identify the number of terms in the series

In this case, \(i\) varies from 0 to 7. So the number of terms \(n\) is 7 - 0 + 1 = 8.
03

Apply the formula for the sum of an arithmetic series

The sum \(s\) of an arithmetic series is given by the formula \[s = n/2 (a+l)\]. Here, \(a=2\), \(l=30\), and \(n=8\). By inserting these values into the formula, we get the sum \(s=8/2*(2+30)=128\).

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Key Concepts

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

Sum of Series
When dealing with series, one common task is to find their sum. A series is basically the result of adding the terms of a sequence, one after the other. For an arithmetic series, where each term is derived by adding a constant to the previous one, there is a specific approach you can use to find its sum.
  • The sum of an arithmetic series can be found using a special formula.
  • The important aspects to identify are the first term, the last term, and the number of terms within the series.
Once you have these values, you can simply plug them into the series formula. Terms behave like puzzle pieces that fit together to make the complete sum. Understanding how to put those pieces together is crucial for solving series-based problems.
Arithmetic Sequence
An arithmetic sequence is a series of numbers where each number after the first is the previous number plus a constant difference, known as the common difference.
  • This common difference is key to the sequence, determining how each term grows.
  • In our case, the sequence generated from the expression \( 2 + 4i \) has a first term \( a = 2 \) and a common difference of 4.
Identifying the common difference helps you understand the structure of the sequence. If you know the first term and the common difference, you can find any term in the sequence just by continuing the pattern. This understanding also plays a crucial role when applying sums in arithmetic sequences.
Series Formula
The formula to find the sum of an arithmetic series is a tool that makes calculations straightforward. This formula is expressed as: \[s = \frac{n}{2} (a + l)\]Where:
  • \( s \) is the sum of the series,
  • \( n \) is the total number of terms,
  • \( a \) is the first term,
  • \( l \) is the last term.
In practice, this formula helps quickly get to the solution by focusing only on a few important values. Instead of adding every single term together manually, use the series formula to compute the sum in no time. This method highlights the beauty of pattern recognition and mathematical shortcuts, making problems much easier to handle.

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