/*! 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 Find the first five terms of eac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 25

Find the first five terms of each arithmetic sequence described. Find \(a_{8}\) if \(a_{n}=4+3 n\)

Short Answer

Expert verified
First five terms: 4, 7, 10, 13, 16. Eighth term: 25.

Step by step solution

01

Understand the Arithmetic Sequence Formula

The given formula for the arithmetic sequence is \(a_{n} = 4 + 3n\). This is a formula where \(4\) is the first term \(a_1\) and \(3n\) is the common difference multiplied by \(n\).
02

Calculate the First Term

To find the first term \(a_1\), substitute \(n=0\) into the sequence formula: \(a_1 = 4 + 3(0) = 4\).
03

Calculate Subsequent Terms

For each of the next terms, increase \(n\) by 1. Calculate:- Second term \(a_2 = 4 + 3(1) = 7\)- Third term \(a_3 = 4 + 3(2) = 10\)- Fourth term \(a_4 = 4 + 3(3) = 13\)- Fifth term \(a_5 = 4 + 3(4) = 16\)
04

Calculate the Eighth Term

Substitute \(n=7\) in the sequence formula to find \(a_8\):\(a_8 = 4 + 3(7) = 4 + 21 = 25\).

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.

Sequence Formula
An arithmetic sequence is a list of numbers where each term increases or decreases by the same amount each time. This consistent change is captured using the **sequence formula**. The general form of this formula is given by:\[ a_{n} = a_1 + (n-1)d \]where:
  • \(a_{n}\) is the \(n\)th term.
  • \(a_1\) is the first term of the sequence.
  • \(d\) is the common difference between consecutive terms.
  • \(n\) is the position of the term in the sequence.

In the exercise, we have a specific formula, \(a_{n} = 4 + 3n\). This might look slightly different than the general formula, but it conveys the same idea. The terms are generated using a fixed starting point and then incremented by a common difference.
Common Difference
The **common difference** is the amount we add or subtract to go from one term in the sequence to the next. It remains the same throughout the entire sequence.
In our example formula, \(a_{n} = 4 + 3n\), the common difference, often denoted by \(d\), is the number that we multiply by \(n\). Here, the common difference is clearly \(3\). This means every term is 3 more than the previous term.
Understanding the common difference is crucial, as it determines how the sequence progresses. It tells us how much each term increases, helping us to easily predict or calculate any term in the sequence.
First Term
The **first term** in an arithmetic sequence is the starting point of the sequence. It's foundational for calculating all subsequent terms.
In the formula \(a_{n} = 4 + 3n\), our first term, denoted by \(a_1\), can be found by substituting \(n = 0\). Substituting in this way, we get:\[a_1 = 4 + 3(0) = 4\]The first term, \(4\), establishes the beginning of our sequence. With it, and knowing the common difference, any other term can be obtained.
Substitute in Formula
To find specific terms in an arithmetic sequence, you **substitute** the desired term number into the sequence formula.
Let's use the formula from our exercise, \(a_{n} = 4 + 3n\), to find any specific term. For example, if we want to find the first term \(a_1\), we substitute \(n=0\):
  • \(a_1 = 4 + 3(0) = 4\)
For the eighth term, \(a_8\), we substitute \(n=7\):
  • \(a_8 = 4 + 3(7) = 25\)

This substitution process allows you to determine the value of any term in the sequence quickly. It involves replacing \(n\) with the number corresponding to the term's position and performing the simple arithmetic.

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

Prove that each statement is true for all positive integers. $$ 1+2+3+\cdots+n=\frac{n(n+1)}{2} $$

Ofelia is taking ten shots in the intramural free-throw shooting competition. How many sequences of makes and misses are there that result in her making eight shots and missing two?

Find the indicated term of each expansion. fourth term of \(\left(x+\frac{1}{3}\right)^{7}\)

Find the indicated term of each expansion. seventh term of \((x+y)^{12}\)

OPEN ENDED Write a power of a binomial for which the first term of the expansion is 625\(x^{4}\) .
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.