/*! 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 13 Find the 32nd term of each seque... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 13

Find the 32nd term of each sequence. \(0.1,0.5,0.9,1.3, \dots\)

Short Answer

Expert verified
The 32nd term of the sequence is 12.5.

Step by step solution

01

Identify the Common Difference

Determine the common difference(d) of the arithmetic sequence by subtracting the first term from the second term. For the given sequence, it should be \(0.5 - 0.1 = 0.4\)
02

Apply the Arithmetic Sequence Formulas

To find the nth term of an arithmetic sequence, use the formula \(a_n = a_1 + (n-1)*d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number and \(d\) is the common difference.
03

Calculate the 32nd Term

Substitute the values into the equation: \(a_{32} = 0.1 + (32 - 1)*0.4 = 0.1 + 31*0.4 = 0.1 + 12.4 = 12.5\) So, the 32nd term of the sequence is 12.5

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
The common difference is a crucial concept in understanding arithmetic sequences. This is the fixed amount that is added (or subtracted) from one term to the next to maintain the sequence's uniformity. In an arithmetic sequence, each term increases or decreases by this same value every time. To identify the common difference, one simply subtracts any term from the subsequent term in the sequence.
  • For instance, take the sequence: 0.1, 0.5, 0.9, 1.3, ...
  • The common difference is calculated as follows: 0.5 - 0.1 = 0.4
It is critical to ensure any sequence examined follows this pattern across all consecutive terms. This consistency confirms the nature of the sequence as arithmetic.
Nth Term Formula
The nth term formula is a powerful tool for determining any term in an arithmetic sequence without listing all the preceding terms. By using this formula, you can jump straight to the desired term.
The formula is given by: \[ a_n = a_1 + (n-1) \times d \]
  • Where \(a_n\) is the term you want to find,
  • \(a_1\) is the first term of the sequence,
  • \(n\) is the term number, and
  • \(d\) is the common difference.
Plugging these values into the formula allows you to efficiently solve for any term.
It's particularly useful for large term numbers, like finding the 32nd term in our given sequence.
Sequence Pattern
Recognizing patterns in a sequence is essential in understanding how arithmetic sequences progress. A sequence pattern showcases the uniform manner in which terms increase or decrease. For arithmetic sequences, this pattern is driven by the common difference.
For example, in the sequence 0.1, 0.5, 0.9, 1.3, ... each term increases by 0.4, highlighting the arithmetic pattern.
  • Spotting these patterns helps predict future terms.
  • It assures that our calculation and formula are applied correctly.
By understanding the pattern, you develop the ability to create sequences or validate the terms already calculated. It's a fundamental step in mastering 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

Determine whether the sum of each infinite geometric series exists. $$ -972-324-108-\dots $$

Technology Create a spreadsheet to evaluate the first \(n\) terms of each series. Determine whether each infinite series converges to a sum. If so, estimate the sum. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}} $$

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. \(2+4+6+8+\ldots ; n=20\)

Graph each curve. Use inscribed rectangles to approximate the area under the curve for the interval and rectangle width given. $$ y=x^{3}, 1 \leq x \leq 3,0.25 $$

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. \(a_{n}=3 n(n+1)\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.