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Chapter 14: Problem 46

Write each series using summation notation. $$ 7+8+9+10+11 $$

Short Answer

Expert verified
\[\sum_{n=1}^{5}(6+n)\]

Step by step solution

01

Identify the Pattern

First, observe that the given series \[7 + 8 + 9 + 10 + 11\] increments by 1 in each subsequent term. This signifies an arithmetic series.
02

Determine General Term

In an arithmetic series, each term can be expressed as \[a_n = a_1 + (n-1)d\]. Here, the first term (\(a_1\)) is 7 and the common difference (\(d\)) is 1. So the general term, \(a_n\), can be written as:\[a_n = 7 + (n-1)\cdot1 = 7 + n - 1 = 6 + n.\]
03

Identify the Range

The given series has 5 terms, starting from \(n = 1\) to \(n = 5\). Thus, the range for summation notation is from 1 to 5.
04

Write in Summation Notation

Combine all the information and write the series in summation notation:\[\sum_{n=1}^{5} (6 + n)\]

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Key Concepts

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

arithmetic series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference. In the given exercise, the series is 7 + 8 + 9 + 10 + 11. Here, each number increases by 1 from the previous one, meaning the common difference (d) is 1.
The general formula for an arithmetic series is:
  • The first term ( a_1e
  • The common difference (den)
  • The general term ( a_ne)
    • Enth term
Understanding this concept is crucial for later steps, such as writing the series in summation notation.
general term
In any arithmetic series, you can find the general term, a_n, using the formula:
  • a_n = a_1 + (n - 1)d.
In our example, where a_1 = 7en, d = 1e. The general term is calculated as follows:
  • a_n = 7 + (n - 1) e leads to
    This simplifies to:
    • a_n = 6 + n.

      • By understanding how to determine the general term, you can easily identify any term in the series and later use this in summation notation.
series summation
Once you have the general term, you can express the series in summation notation.Summation notation provides a concise way to express the sum of a sequence of terms. For the given series of 5 terms, the general term is a_n = 6 + n.Summation notation has the form:
  • says the lower limit of the index is 1, and it goes up to the upper limit, which is 5.

    • Therefore, the series can be written in summation notation as:
    • _{n=1}^{5} (6 + n)

    This concise form makes it clear that you are summing the terms from n=1 to n=5.

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Most popular questions from this chapter

Find the sum, if it exists, of the terms of each infinite geometric sequence. See Examples 8 and 9. $$ \sum_{i=1}^{\infty} \frac{3}{5}\left(\frac{5}{6}\right)^{i} $$

Write the first five terms of each arithmetic sequence. \(a_{1}=-2, d=-4\)

Solve each application. When dropped from a certain height, a ball rebounds to \(\frac{3}{5}\) of the original height. How high will the ball rebound after the fourth bounce if it was dropped from a height of \(10 \mathrm{ft}\) ? Round to the nearest tenth.

Solve each application. A tracer dye is injected into a system with an ingestion and an excretion. After \(1 \mathrm{hr}, \frac{2}{3}\) of the dye is left. At the end of the second hour, \(\frac{2}{3}\) of the remaining dye is left, and so on. If one unit of the dye is injected, how much is left after \(6 \mathrm{hr} ?\)

Write the first five terms of each sequence. $$ a_{n}=6(-1)^{n+1} $$
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