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Chapter 14: Problem 9

Write the first five terms of each arithmetic sequence. $$ a_{1}=-2, d=-4 $$

Short Answer

Expert verified
The first five terms are -2, -6, -10, -14, -18.

Step by step solution

01

Identify the first term and common difference

The first term of the arithmetic sequence is given as \(a_{1} = -2\). The common difference is given as \(d = -4\).
02

Calculate the second term

To find the second term \(a_{2}\), add the common difference \(d\) to the first term \(a_{1}\).\ \a_{2} = a_{1} + d = -2 + (-4) = -6\.
03

Calculate the third term

To find the third term \(a_{3}\), add the common difference \(d\) to the second term \(a_{2}\).\ \a_{3} = a_{2} + d = -6 + (-4) = -10\.
04

Calculate the fourth term

To find the fourth term \(a_{4}\), add the common difference \(d\) to the third term \(a_{3}\).\ \a_{4} = a_{3} + d = -10 + (-4) = -14\.
05

Calculate the fifth term

To find the fifth term \(a_{5}\), add the common difference \(d\) to the fourth term \(a_{4}\).\ \a_{5} = a_{4} + d = -14 + (-4) = -18\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

first term
The first term in an arithmetic sequence is crucial because it serves as the starting point for the sequence.
In this exercise, the first term is given as \(a_{1} = -2\).
This means the sequence begins at -2.
No matter how many steps we calculate in the sequence, we always begin here.
The first term helps us establish a reference point for all subsequent terms in the sequence.
Without it, we wouldn't know where to start.
common difference
The common difference, denoted as \(d\), is a key component of arithmetic sequences.
It tells us how much each term increases or decreases from the previous term.
In this problem, the common difference is \(d = -4\).
This means each term in the sequence is 4 less than the term before it.
Understanding the common difference is essential for predicting future terms.
sequence calculation
Sequence calculation in an arithmetic sequence involves using the first term and the common difference to find subsequent terms.
To find the second term \(a_{2}\), we add the common difference to the first term:
\(a_{2} = a_{1} + d = -2 + (-4) = -6\).
To find the third term \(a_{3}\), we add the common difference to the second term:
\(a_{3} = a_{2} + d = -6 + (-4) = -10\).
This calculation process is repeated for each new term.
arithmetic progression
Arithmetic progression is another term for an arithmetic sequence.
An arithmetic sequence follows a predictable pattern of adding the same value (the common difference) to each term to get the next term.
For example, starting from -2 and repeatedly adding -4, we get:
-2, -6, -10, -14, -18.
This clear pattern and repetition make arithmetic sequences powerful tools for understanding and predicting numerical relationships.

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Most popular questions from this chapter

Solve each problem involving an ordinary annuity. B. G. Thompson puts \(\$ 1000\) in a retirement account at the end of each quarter \(\left(\frac{1}{4} \text { of a year }\right)\) for 15 yr. If the account pays \(4 \%\) annual interest compounded quarterly, how much will be in the account at that time?

Solve each applied problem by writing the first few terms of a sequence. Leslie Maruri is offered a new modeling job with a salary of \(20,000+2500 n\) dollars per year at the end of the \(n\) th year. Write a sequence showing her salary at the end of each of the first 5 yr. If she continues in this way, what will her salary be at the end of the tenth year?

Find a general term \(a_{n}\) for the given terms of each sequence. $$ -10,-20,-30,-40, \dots $$

Find the arithmetic mean for each collection of numbers. $$ 5,9,8,2,4,7,3,2,0 $$

Write out the first five terms of each sequence. $$ a_{n}=n+4 $$
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