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Chapter 14: Problem 78

Rewrite each sum using sigma notation. Answers may vary. $$ 7+14+21+28+35+\cdots $$

Short Answer

Expert verified
\( \sum_{n=1}^{k} 7n \)

Step by step solution

01

Identify the Pattern

First, notice that each term in the sum increases by 7 each time. This tells us we have an arithmetic sequence where the common difference is 7.
02

Find the General Term

In an arithmetic sequence, the general term can be found using the formula: \[ a_n = a_1 + (n-1)d \] Where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 7\) and \(d = 7\). So, \[ a_n = 7 + (n-1) \times 7 \] which simplifies to \[ a_n = 7n \]
03

Write in Sigma Notation

The sum can be written using sigma notation as: \[ \sum_{n=1}^{k} 7n \] where \(k\) is the number of terms we want to sum up.

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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 list of numbers where each term increases (or decreases) by a constant amount. This constant amount is called the 'common difference.' In the given exercise, the sequence begins with 7 and the common difference is also 7. An arithmetic sequence can be written as:
  • First term: 7
  • Second term: 14
  • Third term: 21
  • and so on...
To identify an arithmetic sequence:
  • Look at the difference between consecutive terms.
  • If the difference remains constant, it's an arithmetic sequence.
Common Difference
The 'common difference' in an arithmetic sequence is the amount added (or subtracted) to go from one term to the next. In the given sequence -- 7, 14, 21, 28, 35, ... -- the common difference is 7. You can find this by:
  • Subtracting any term from the next term.
  • For example: 14 - 7 = 7, 21 - 14 = 7, and so on.
The common difference can be positive (increasing sequence) or negative (decreasing sequence). Understanding the common difference helps in identifying the pattern and is crucial for finding the general term.
General Term
The general term of an arithmetic sequence is a formula that allows you to find any term in the sequence. It is represented as: a_n = a_1 + (n-1)d where
  • a_n is the nth term
  • a_1 is the first term
  • d is the common difference
In the given example, the first term a_1 is 7 and the common difference d is 7. Plugging these values into the formula, we get:
  • a_n = 7 + (n-1) * 7
Simplifying this, we find:

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Most popular questions from this chapter

The sequence \(1,4,9,16, \ldots\) can be written as \(f(x)=x^{2}\) with the domain the set of all positive integers. Explain how the graph of \(f\) would compare with the graph of \(y=x^{2}\)

Review finding equations. Find an equation of the line satisfying the given conditions. Containing the point \((-1,-4)\) and perpendicular to the line given by \(3 x-4 y=7[3.7]\)

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Write out and evaluate each sum. $$ \sum_{k=0}^{5}\left(k^{2}-3 k+4\right) $$
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