/*! 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 25 Determine an expression for the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 25

Determine an expression for the general term \(a_{n}\) of each sequence \(4,8,12,16, \ldots\)

Short Answer

Expert verified
The general term of the sequence is \(a_{n} = 4n\).

Step by step solution

01

- Identify the First Term

The first term of the sequence is given as 4. Denote this by: \(a_1 = 4\)
02

- Determine the Common Difference

Find the difference between consecutive terms in the sequence: \(8 - 4 = 4\), \(12 - 8 = 4\). The common difference is 4. Denote this by: \(d = 4\)
03

- Use the Formula for the nth Term

The general form of an arithmetic sequence is given by: \(a_{n} = a_1 + (n-1)d\)
04

- Substitute the Values

Substitute the values of the first term \(a_1\) and the common difference \(d\) into the general formula: \(a_{n} = 4 + (n-1) \times 4\)
05

- Simplify the Expression

Simplify the right-hand side of the equation: \(a_{n} = 4 + 4n - 4\) Combine like terms: \(a_{n} = 4n\)

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.

general term
In arithmetic sequences, the **general term** represents any term in the sequence. It allows us to find the value at any position without listing all preceding terms. This is quite useful for large sequences. For example, for the sequence given in the exercise, the general term formula lets us find the 50th term directly. This saves us from manually calculating each term up to the 50th.
common difference
The **common difference** is the consistent difference between consecutive terms in an arithmetic sequence. To find it, simply subtract any term from the term that directly follows it.

The common difference in the given sequence is found as follows:
  • For the first two terms: \(8 - 4 = 4\)
  • For the next terms: \(12 - 8 = 4\)

This shows that the common difference is 4.

Knowing the common difference helps in creating a general term formula.
nth term formula
The **nth term formula** in arithmetic sequences is used to find the value of any term in the sequence. The general formula is:

\[a_{n} = a_1 + (n-1)d\]

Where:
  • \(a_{n}\) is the nth term
  • \(a_1\) is the first term
  • \(d\) is the common difference
  • \(n\) is the position of the term in the sequence

Using the given sequence, we substitute the values:

\[a_{n} = 4 + (n-1) \times 4\]

This formula helps easily find any term without listing the entire sequence.
sequence simplification
After substituting the values in the nth term formula, we often need to **simplify the sequence** expression to make it more understandable or easier to use.

For the given sequence, starting from the formula:

\[a_{n} = 4 + (n-1) \times 4\]
We simplify it by first expanding the expression:

\[a_{n} = 4 + 4n - 4\]
Then, we combine the like terms:

\[a_{n} = 4n\]
This final simplified form allows quick computation of any term's value based on its position.

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

Evaluate the sum of the terms of each geometric sequence. In Exercises 39-44, give answers to the nearest thousandth. See Examples 5 and 6. $$ \sum_{i=1}^{6}(-2)\left(-\frac{1}{2}\right)^{i} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ \frac{5}{7}, \frac{8}{7}, \frac{11}{7}, 2, \ldots $$

Write the first five terms of each arithmetic sequence. \(a_{1}=0.25, d=0.55\)

Evaluate each expression. $$ 4 ! $$

Determine an expression for the general term \(a_{n}\) of each sequence \(-8,-16,-24,-32, \ldots\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.