/*! 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 65 Find the 27 th term of each sequ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 65

Find the 27 th term of each sequence. $$ 1,3.5,6,8.5, \dots $$

Short Answer

Expert verified
The 27th term of the given arithmetic sequence is 66.

Step by step solution

01

Identify the Common Difference

The given sequence is 1, 3.5, 6, 8.5, and so on. By subtracting the first term from the second, and the second from the third, a common difference of 2.5 is discovered.
02

Use the Formula for the General Term

The formula for the nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, d is the common difference, and n is the position of the term in the sequence.
03

Insert Values into the Formula

We can use the formula, substituting the known values into it. Hence, \( a_27 = 1 + (27-1) * 2.5 \)
04

Calculate

Finally, we can calculate the value. It gives \( a_27 = 1 + 26 * 2.5 = 1 + 65 = 66 \)

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.

Common Difference
In an arithmetic sequence, the common difference is a key component that defines the sequence. It is the consistent amount added (or subtracted) from one term to the next to generate the sequence. In the sequence given in the exercise: 1, 3.5, 6, 8.5, ..., you can find the common difference by subtracting the first term from the second term, and the second term from the third term, and so on. For example:
  • 3.5 - 1 = 2.5
  • 6 - 3.5 = 2.5
  • 8.5 - 6 = 2.5
This arithmetic difference of 2.5 is consistent between all terms, confirming it to be the common difference. Understanding the common difference is crucial as it allows us to predict future terms in a sequence.
General Term Formula
The general term formula is the mathematical expression that represents the nth term of an arithmetic sequence. It is expressed as:\[ a_n = a_1 + (n-1)d \]Where:
  • \( a_n \) is the nth term we are trying to find
  • \( a_1 \) is the first term of the sequence
  • \( d \) is the common difference
  • \( n \) is the term number or position in the sequence
Using this formula, you can determine any term in an arithmetic sequence without listing all previous terms. This formula is essential because it allows you to quickly find terms even if they are far into the sequence.
nth Term Calculation
To calculate the nth term in the sequence, you apply the values you already know into the general term formula. Let's use our sequence example here, where we need to find the 27th term.Substitute the known values into the formula:
  • \( a_1 = 1 \) (the first term)
  • \( d = 2.5 \) (the common difference)
  • \( n = 27 \) (the term number we are calculating)
Applying these values to the general term formula:\[ a_n = 1 + (27-1) \times 2.5 \]Therefore, \( a_{27} = 1 + 26 \times 2.5 \), which simplifies to:\[ a_{27} = 1 + 65 = 66 \]This calculation shows the 27th term of the sequence is 66. Understanding this process is valuable for using arithmetic sequences in real-world scenarios, like finance and statistics.

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

For Exercises \(79-82,\) suppose tan \(\theta=\frac{4}{3}, \sin \theta>0,\) and \(-\frac{\pi}{2} \leq \theta<\frac{\pi}{2}\) . Enter each answer as a decimal. What is \((\sin \theta)(\cot \theta) ?\)

Evaluate each expression. Give your answer as a decimal rounded to the nearest hundredth. $$ \sec 200^{\circ} $$

Graph each function in the interval from 0 to 2\(\pi .\) Describe any phase shift and vertical shift in the graph. $$ y=-2 \sec (x-4) $$

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \cot 60^{\circ} $$

Evaluate each expression. Give your answer as a decimal rounded to the nearest hundredth. $$ \csc 80^{\circ} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

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.