/*! 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 19 Evaluate the indicated term for ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 19

Evaluate the indicated term for each arithmetic sequence. $$ 2,4,6, \ldots ; \quad a_{24} $$

Short Answer

Expert verified
The 24th term is 48.

Step by step solution

01

- Identify the first term (a)

In an arithmetic sequence, the first term is denoted by \(a\). From the given sequence, the first term is 2.
02

- Identify the common difference (d)

The common difference \(d\) is the difference between any two consecutive terms. From the given sequence, \(d = 4 - 2 = 2\).
03

- Use the general formula for the nth term

The general formula for the nth term of an arithmetic sequence is \(a_n = a + (n-1)d\).
04

- Plug in the values to find the 24th term

We need the 24th term \(a_{24}\). Using the values identified earlier: \[ a_{24} = 2 + (24-1) \times 2 \] Simplify: \[ a_{24} = 2 + 46 = 48 \]

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 any arithmetic sequence, the first term is denoted by the letter \(a\). It's the starting point of the sequence. Let's look at the given sequence: \(2, 4, 6, \ldots\). Here, the first term \(a\) is clearly 2. Understanding the first term is crucial because it sets the foundation for the entire sequence. Every subsequent number in the sequence builds upon this initial value. Whenever you're asked to find specific terms in an arithmetic sequence, always start by identifying this first term.
common difference
The common difference, denoted as \(d\), plays a key role in arithmetic sequences. It's the difference between any two consecutive terms. To find \(d\), you simply subtract the first term from the second term. For our sequence \(2, 4, 6, \ldots\), we calculate the common difference as follows:
  • 4 - 2 = 2
  • 6 - 4 = 2

As you can see, the common difference \(d\) in this sequence is 2. This value remains constant throughout the entire sequence, allowing us to predict future terms easily.
nth term formula
To find any term in an arithmetic sequence, you can use the nth term formula. The formula is:
\[ a_n = a + (n-1)d \]
This formula helps you determine the value of any term based on its position \(n\), the first term \(a\), and the common difference \(d\). Let's decode this formula step by step:
  • \(a_n\) is the term you want to find.
  • \(a\) is the first term.
  • \(n\) is the position of the term in the sequence.
  • \(d\) is the common difference.

For the 24th term \(a_{24}\) in our sequence, substitute the values of \(a\), \(d\), and \(n\) into the formula: \[ a_{24} = 2 + (24-1) \times 2 \].
sequence evaluation
When evaluating a specific term in an arithmetic sequence, all you've learned comes together. For the given sequence: \(2, 4, 6, \ldots\), you're asked to find the 24th term \(a_{24}\). Using the formula \(a_n = a + (n-1)d\):
  • Substitute \(a = 2\) (first term)
  • Substitute \(d = 2\) (common difference)
  • Substitute \(n = 24\) (term position)

This gives us:
\[ a_{24} = 2 + (24-1) \times 2 \]
Simplify the expression:
\[ a_{24} = 2 + 46 = 48 \]
Therefore, the 24th term in the sequence is 48. By following these steps, you can evaluate any term in an arithmetic sequence accurately.

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

Find the sum, if it exists, of the terms of each infinite geometric sequence. $$ a_{1}=6, r=\frac{1}{3} $$

Find a general term for each geometric sequence. $$ -3, \frac{3}{2},-\frac{3}{4}, \ldots $$

Use a formula for \(S_{n}\) to evaluate each series. $$ \sum_{i=1}^{20}\left(\frac{3}{2} i+4\right) $$

Find the arithmetic mean for each collection of numbers. $$ 8,11,14,9,7,6,8 $$

Multiply. $$ (3 x+2 y)^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.