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Chapter 14: Problem 30

Use a graphing calculator to find the first 5 terms of each sequence. $$ a_{n}=8 n-17 $$

Short Answer

Expert verified
-9, -1, 7, 15, 23

Step by step solution

01

Understand the sequence formula

The sequence is given by the formula \(a_{n} = 8n - 17\), where \(n\) represents the term number.
02

Identify the first 5 term numbers

The first 5 term numbers for the given sequence are \(n = 1, 2, 3, 4, 5\).
03

Calculate the first term \(a_1\)

Substitute \(n = 1\) into the formula: \(a_{1} = 8(1) - 17 = -9\).
04

Calculate the second term \(a_2\)

Substitute \(n = 2\) into the formula: \(a_{2} = 8(2) - 17 = -1\).
05

Calculate the third term \(a_3\)

Substitute \(n = 3\) into the formula: \(a_{3} = 8(3) - 17 = 7\).
06

Calculate the fourth term \(a_4\)

Substitute \(n = 4\) into the formula: \(a_{4} = 8(4) - 17 = 15\).
07

Calculate the fifth term \(a_5\)

Substitute \(n = 5\) into the formula: \(a_{5} = 8(5) - 17 = 23\).

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Key Concepts

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

sequence formula
An arithmetic sequence is a number pattern where each term after the first is obtained by adding a fixed, constant number, called the common difference, to the previous term. In the given formula, a_n = 8n - 17, the sequence is defined by the relationship between the term number ( n) and the actual numerical value of the term ( a_n ).
Understanding how to use and manipulate the sequence formula is key to performing other operations, like finding specific terms in the sequence or generating new sequences from a given rule.
term calculation
To find the first 5 terms of our arithmetic sequence, a_n = 8n - 17, we need to substitute the term numbers ( n = 1, 2, 3, 4, 5) into the sequence formula.
- For the first term ( n = 1): a_1 = 8(1) - 17 = -9.
- For the second term ( n = 2): a_2 = 8(2) - 17 = -1.
- For the third term ( n = 3): a_3 = 8(3) - 17 = 7.
- For the fourth term ( n = 4): a_4 = 8(4) - 17 = 15.
- For the fifth term ( n = 5): a_5 = 8(5) - 17 = 23.
Calculating each term involves substituting the term number into the formula and simplifying the expression. It's straightforward once you understand the pattern.
graphing calculator
A graphing calculator is a useful tool for visualizing arithmetic sequences and analyzing their behavior. To find the first 5 terms of the sequence a_n = 8n - 17, you can follow these steps on a graphing calculator:
  • Input the sequence formula into the calculator's function mode.
  • Set the variable to increment based on your term numbers (1, 2, 3, 4, 5).
  • Evaluate the function at each increment to find the corresponding term values (-9, -1, 7, 15, 23).
Graphing those points can help you see the linear nature of arithmetic sequences. The graph will show a straight line if plotted correctly, indicating a constant rate of change, which is a characteristic of arithmetic sequences.

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Most popular questions from this chapter

Determine whether each infinite geometric series has a limit. If a limit exists, find it. $$ 0.43+0.0043+0.000043+\cdots $$

Write out and evaluate each sum. $$ \sum_{k=1}^{5} \frac{1}{2 k} $$

Find the nth, or general, term for each geometric sequence. $$ 2,4,8, \dots $$

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=1, a_{n+1}=5 a_{n}-2 $$

Explain why the equation $$ \sum_{k=1}^{n}\left(a_{k}+b_{k}\right)=\sum_{k=1}^{n} a_{k}+\sum_{k=1}^{n} b_{k} $$ is true for any positive integer \(n .\) What laws are used to justify this result?
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