/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Find the first term and the comm... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 11

Find the first term and the common difference. $$ 7,3,-1,-5, \ldots $$

Short Answer

Expert verified
First term: 7, Common difference: -4.

Step by step solution

01

Identify the first term

The first term of the sequence is the initial number. Here, the first term is 7.
02

Determine the common difference

The common difference is found by subtracting the first term from the second term. In this case, \(3 - 7 = -4\). To verify, check the difference between subsequent terms: \(-1 - 3 = -4\) and \(-5 - (-1) = -4\). Thus, the common difference is -4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

First Term
In arithmetic sequences, the first term is crucial. It sets the stage for all following terms. The first term, denoted usually as \(a_1\), is the initial number in the sequence.
In our example:
7, 3, -1, -5, ...
The first term is 7.
This means the sequence starts with 7.
Once the first term is defined, it allows us to systematically calculate the subsequent terms using the common difference.
Common Difference
The common difference is essential for arithmetic sequences as it represents the constant interval between successive terms. To find it, subtract the first term (a1) from the second term (a2).
For example:
In the given sequence 7, 3, -1, -5, ... we calculate:
\(3 - 7 = -4\)
To ensure consistency, check the difference between other consecutive terms:
  • \(-1 - 3 = -4\)
  • \(-5 - (-1) = -4\)
Therefore, the common difference, denoted usually as \(d\), is -4.
Understanding the common difference helps in predicting and computing future terms.
Subsequent Terms
After identifying the first term and the common difference, you can easily determine the subsequent terms. Each term in an arithmetic sequence is obtained by adding the common difference to the previous term.
Using our sequence 7, 3, -1, -5, ... and the common difference of -4:
  • First term, \(a_1 = 7\)
  • Second term, \(a_2 = 7 + (-4) = 3\)
  • Third term, \(a_3 = 3 + (-4) = -1\)
  • Fourth term, \(a_4 = -1 + (-4) = -5\)
This pattern continues indefinitely.
Knowing how to calculate subsequent terms is helpful for solving problems and understanding the behavior of arithmetic sequences.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A superball dropped from the top of the Washington Monument ( 556 ft high) rebounds three-fourths of the distance fallen. How far (up and down) will the ball have traveled when it hits the ground for the 6 th time?

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 \(y=x^{2}\)

Classify each of the following as an arithmetic sequence, a geometric sequence, an arithmetic series, a geometric series, or none of these. $$ 2,6,18,54, \dots $$

Find the first term and the common difference. $$ \$ 5.12, \$ 5.24, \$ 5.36, \$ 5.48, \ldots $$

The nth term of a sequence is given. In each case, find the first 4 terms, the 10 th term, \(a_{10},\) and the 15 th term, \(a_{15},\) of the sequence. $$ a_{n}=5 n-2 $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.