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Chapter 15: Problem 14

Write the sum using sigma notation. $$ 7+10+13+16+19+22 $$

Short Answer

Expert verified
Question: Represent the sum of the arithmetic sequence 7, 10, 13, 16, 19, and 22 in sigma notation. Answer: $$\sum_{n=1}^{6} (7 + (n - 1)3)$$

Step by step solution

01

Identify the starting point and the common difference

The first term of the sequence is 7, and the common difference between each term is 3 (e.g., 10 - 7 = 3, 13 - 10 = 3, etc.).
02

Determine the number of terms in the sequence

Count the number of terms given: 7, 10, 13, 16, 19, and 22. There are a total of 6 terms in this sequence.
03

Write the arithmetic sequence formula

An arithmetic sequence can be represented by the formula $$a_n = a_1 + (n - 1)d$$, where $$a_n$$ is the nth term, $$a_1$$ is the first term, n is the position of the term, and d is the common difference.
04

Express the sum using sigma notation

Using sigma notation, the sum of the arithmetic sequence can be represented as: $$ \sum_{n=1}^{6} a_n = \sum_{n=1}^{6} (7 + (n - 1)3) $$ Now, the given exercise is written in sigma notation.

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Key Concepts

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

Understanding an Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms is consistent. This predictable pattern makes it a fundamental concept in mathematics. For example, in a sequence like 7, 10, 13, 16, 19, and 22, each number increases by the same amount as the previous one.
In this sequence, 7 is the first term, and it is followed by each subsequent term created by adding a fixed number to it. The beauty of an arithmetic sequence lies in its regularity, which helps in solving problems and finding unknown terms.
  • The first term is denoted as \( a_1 \) in an arithmetic sequence.
  • Each subsequent term can be found by adding a constant number to the previous term.
  • This constant number is known as the "common difference".
Understanding these basic elements is essential for working with arithmetic sequences.
What is the Common Difference?
The common difference is a key feature of an arithmetic sequence that defines the change between consecutive terms. It is what makes the sequence arithmetic and not just a random set of numbers.
To find the common difference in a sequence, subtract any term from the term that follows it. In our example, the sequence is 7, 10, 13, 16, 19, and 22.
  • Name the common difference as \( d \).
  • In our sequence, subtract 7 from 10, 10 from 13, and so on: each gives us 3, hence \( d = 3 \).
  • This consistency across the entire sequence confirms its arithmetic nature.
The common difference significantly impacts the behavior of the sequence and allows us to construct a formula to find any term in that sequence.
Understanding the Arithmetic Sequence Formula
The arithmetic sequence formula is a tool that allows us to find any term in the sequence without listing all the terms. It is represented as \( a_n = a_1 + (n - 1)d \). This formula can be broken down into simple components:
  • \( a_n \) is the nth term you wish to find.
  • \( a_1 \) represents the first term in the sequence, which is the starting point.
  • \( n \) is the position of the desired term within the sequence.
  • \( d \) is the common difference, which is added to the previous term.
For example, if you want to find the 6th term of our sequence, you plug the first term (7), common difference (3), and the term position (6) into the formula, giving \( a_6 = 7 + (6-1) \times 3 = 7 + 15 = 22 \). This method simplifies the process of working with and identifying terms in any arithmetic sequence.

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Most popular questions from this chapter

Find the sum of the series. $$ 1 / 81+1 / 27+1 / 9+\cdots+243 $$

Are the sequences geometric? For those that are, give a formula for the \(n^{\text {th }}\) term. $$ -2,4,-8,16, \ldots $$

In Exercises \(12-15\), write the first 5 terms of the sequence \(a_{n}=f(n)\) $$ f(t)=3 t-1 $$

Find the sum of the series. $$ \sum_{k=0}^{6} 2\left(\frac{3}{4}\right)^{k} $$

(a) Use sigma notation to write the sum \(2+4+6+\) \(\cdots+18\) (b) Use the arithmetic series sum formula to find the sum in part (a).
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