/*! 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 17 Find the 15 th term of the arith... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 17

Find the 15 th term of the arithmetic sequence \(7,10,13, \ldots\)

Short Answer

Expert verified
The 15th term is 49.

Step by step solution

01

- Identify the first term and the common difference

In the given arithmetic sequence, the first term (\(a\)) is 7 and the common difference (\(d\)) is the difference between the second term and the first term, which is 10 - 7 = 3.
02

- Use the formula for the n-th term of an arithmetic sequence

The formula for the n-th term (\(a_n\)) of an arithmetic sequence is \[a_n = a + (n-1) \times d\].
03

- Substitute the known values

Substitute \(a = 7\), \(d = 3\), and \(n = 15\) into the formula: \[a_{15} = 7 + (15-1) \times 3\].
04

- Simplify the expression

Calculate \(15 - 1\) to obtain 14. Then multiply: \(14 \times 3 = 42\). Finally add this to the first term: \(7 + 42 = 49\).

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 any arithmetic sequence, each term after the first is obtained by adding a constant to the previous term. This constant is known as the common difference, denoted by \(d\).
To find the common difference, subtract the first term from the second term. For example, in the sequence \(7, 10, 13, \ldots\), the common difference \(d\) is calculated as follows:

\[d = 10 - 7 = 3\] The common difference can be positive, negative, or zero. A positive common difference means the terms increase, while a negative common difference means they decrease.
Understanding the common difference is crucial because it is used in the formula for finding any term in the sequence.
n-th term formula
The n-th term formula is used to find any term in an arithmetic sequence without listing all terms up to that point. The formula is as follows:

\[a_n = a + (n-1) \times d\]
Here, \(a_n\) represents the n-th term, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number you want to find.
This formula simplifies finding any term in the sequence. For example, to find the 15th term in the sequence \(7, 10, 13, \ldots\), you would use:

\[a_{15} = 7 + (15-1) \times 3\]
By knowing this formula, you can quickly pinpoint any term in the sequence without endlessly counting.
sequence term calculation
Calculating the terms in a sequence involves applying the common difference and n-th term formula. Let鈥檚 break down the process using our example: finding the 15th term of the sequence \(7, 10, 13, \ldots\).
  • First, identify the first term \(a = 7\) and the common difference \(d = 3\).
  • Next, determine the term number you need, which is 15 in this case.
  • Plug these values into the formula:

\[a_{15} = 7 + (15-1) \times 3\]
Now, simplify inside the brackets:
\[15 - 1 = 14\]
Then, multiply:
\[14 \times 3 = 42\]
Finally, add this to the first term:
\[7 + 42 = 49\]
Thus, the 15th term of the sequence is 49.
By following these steps, you can find any term in an arithmetic sequence easily.

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

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=0, a_{n+1}=a_{n}^{2}+3 $$

Use the formula for \(S_{n}\) to find the indicated sum for each geometric series. $$ S_{16} \text { for } \$ 200+\$ 200(1.06)+\$ 200(1.06)^{2}+\cdots $$

Determine whether each infinite geometric series has a limit. If a limit exists, find it. $$ 0.37+0.0037+0.000037+\cdots $$

Explain how someone can determine the \(x^{2}\) -term of the expansion of \(\left(x-\frac{3}{x}\right)^{10}\) without calculating any other terms.

Find fraction notation for each infinite sum. Each can be regarded as an infinite geometric series. $$ 0.15151515 \ldots $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.