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Chapter 14: Problem 77

Rewrite each sum using sigma notation. Answers may vary. $$ 5+10+15+20+25+\cdots $$

Short Answer

Expert verified
\(\sum_{n=1}^{\infty} 5n\)

Step by step solution

01

- Identify the sequence

Analyze the given sum and recognize the pattern. The sequence given is: \(5, 10, 15, 20, 25, \ldots\).
02

- Determine the common difference

Check the difference between consecutive terms. Here, each term increases by 5: \(10 - 5 = 5\), \(15 - 10 = 5\), and so on. So the common difference is 5.
03

- Write the general term

Use the first term and the common difference to write the general term \(a_n\) of the arithmetic sequence. The first term (\(a_1\)) is 5, and the common difference (\(d\)) is 5: \(a_n = 5 + (n-1) \cdot 5\).
04

- Simplify the general term

Simplify the expression for the general term: \(a_n = 5n\).
05

- Write the sum in sigma notation

Express the given sequence in sigma notation. Since the general term is \(5n\), the sum can be written as: \(\sum_{n=1}^{\infty} 5n\).

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Key Concepts

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

arithmetic sequence
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the 'common difference'. An example of an arithmetic sequence is the series given in the exercise: 5, 10, 15, 20, 25, ...
Each term increases by a fixed value of 5. Understanding arithmetic sequences is crucial for recognizing patterns in numbers and summing series.
common difference
The common difference in an arithmetic sequence is the fixed amount that each term increases (or decreases) by. To find the common difference, simply subtract any term from the term that follows it. In our sequence:
  • 10 - 5 = 5
  • 15 - 10 = 5
  • 20 - 15 = 5
So, the common difference (\(d\)) in this example is 5. Recognizing the common difference helps in writing the general term for the sequence.
general term
The general term of an arithmetic sequence allows you to find any term in the sequence without listing all previous terms. It is usually represented as \(a_n\), where \(n\) is the term number.
The formula to calculate the general term is:
\[ a_n = a_1 + (n-1) \times d \]In our example:
  • \(a_1\) (first term) = 5
  • \(d\) (common difference) = 5
So, the general term \(a_n\) is:

\[ a_n = 5 + (n-1) \times 5 = 5 + 5n - 5 = 5n \]Simplifying, we get the expression \(a_n = 5n\).
infinite series
An infinite series is the sum of an infinite sequence of terms. In this exercise, we aimed to rewrite the series using sigma notation, which indicates a sum. Sigma notation uses the symbol \(\sum_{}\) to represent the summation process. For our arithmetic sequence, since the general term is \(5n\), the sum can be expressed as:
\[\sum_{=1}^{\binfty}5n\]
This compact form makes it easier to work with potentially long or infinite sums in mathematics.

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