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Chapter 12: Problem 34

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$15,12.3,9.6,6.9, \dots$$

Short Answer

Expert verified
Common difference: -2.7; Fifth term: 4.2; nth term: \( a_n = 17.7 - 2.7n \); 100th term: -252.3.

Step by step solution

01

Identify the first term

The first term of the arithmetic sequence is the initial value given. In this sequence, the first term is 15.
02

Find the common difference

To find the common difference of the sequence, subtract the first term from the second term: \[ 12.3 - 15 = -2.7 \] Thus, the common difference is -2.7.
03

Calculate the fifth term

The formula for the fifth term of an arithmetic sequence is: \[ a_5 = a_1 + 4d \] Substituting the known values gives: \[ a_5 = 15 + 4(-2.7) = 15 - 10.8 = 4.2 \]Thus, the fifth term of the sequence is 4.2.
04

Determine the formula for the nth term

The formula for the \( n \)th term of an arithmetic sequence is: \[ a_n = a_1 + (n - 1)d \] Using the first term 15 and common difference -2.7, the formula becomes: \[ a_n = 15 + (n-1)(-2.7) \] Simplifying, we get: \[ a_n = 15 - 2.7n + 2.7 = 17.7 - 2.7n \] Thus, the formula for the \( n \)th term is \( a_n = 17.7 - 2.7n \).
05

Calculate the 100th term

Substitute 100 into the \( n \)th term formula: \[ a_{100} = 17.7 - 2.7 \times 100 \] This gives: \[ a_{100} = 17.7 - 270 = -252.3 \] Therefore, the 100th term is -252.3.

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Key Concepts

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

Common Difference
In an arithmetic sequence, understanding the common difference is crucial. The common difference is the constant amount that each term increases or decreases from the previous term. To find the common difference, simply subtract any term in the sequence from the term that follows it. For example, in the sequence given: 15, 12.3, 9.6, 6.9, we can calculate the common difference as follows:
  • Subtract the first term from the second term: \[ 12.3 - 15 = -2.7 \]
The result, \-2.7\, becomes the common difference for the entire sequence. Each term decreases by this amount as we progress through the sequence. Understanding the common difference is essential because it allows us to predict any term in the sequence. It's the building block for other calculations in arithmetic sequences.
nth Term Formula
The formula for finding the \(n\)th term of an arithmetic sequence is a powerful tool. It allows you to determine the value of any term in the sequence without listing all the previous terms. This is particularly helpful in sequences with a large number of terms. The general formula is: \[ a_n = a_1 + (n - 1)d \] Where:
  • \(a_n\) is the \(n\)th term you're trying to find.
  • \(a_1\) is the first term in the sequence.
  • \(d\) is the common difference.
  • \(n\) is the term number.
Using the initial sequence, substituting known values into the formula provides a specific formula: \[ a_n = 15 + (n-1)(-2.7) \] This simplifies to: \[ a_n = 17.7 - 2.7n \] With this formula, you can calculate any term by plugging in the appropriate value of \(n\). For instance, to find the 100th term, replace \(n\) with 100 and perform the calculations.
Terms of a Sequence
The terms of an arithmetic sequence are found using a systematic approach. Each term is generated by adding the common difference to the previous term, starting from the first term. Let鈥檚 break down how we find specific terms using both the common difference and the \(n\)th term formula.
Starting with the initial term of 15, each subsequent term is derived by subtracting 2.7:
  • The second term: 12.3 \(= 15 - 2.7\)
  • The third term: 9.6 \(= 12.3 - 2.7\)
  • The fourth term: 6.9 \(= 9.6 - 2.7\)
Alternatively, specific terms can be quickly pinpointed using the \(n\)th term formula. For instance, the 5th term can be calculated as follows: \[ a_5 = 17.7 - 2.7 \times 5 = 4.2 \] This dual method of determining the terms ensures flexibility and efficiency in finding any number of terms in a sequence.

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Most popular questions from this chapter

A very patient woman wishes to become a billionaire. She decides to follow a simple scheme: She puts aside 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on, doubling the number of cents each day. How much money will she have at the end of 30 days? How many days will it take this woman to realize her wish?

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$$

The first four terms of a sequence are given. Determine whether these terms can be the terms of an arithmetic sequence, a geometric sequence, or neither. Find the next term if the sequence is arithmetic or geometric. (a) \(5,-3,5,-3, \dots\) (b) \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \dots\) (c) \(\sqrt{3}, 3,3 \sqrt{3}, 9, \ldots\) (d) \(1,-1,1,-1, \ldots\) (e) \(2,-1, \frac{1}{2}, 2, \dots\) (f) \(x-1, x, x+1, x+2, \ldots\) (g) \(-3,-\frac{3}{2}, 0, \frac{3}{2}, \dots\) (h) \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots\)

Population of a City A city was incorporated in 2004 with a population of \(35,000 .\) It is expected that the population will increase at a rate of \(2 \%\) per year. The population \(n\) years after 2004 is given by the sequence $$ P_{n}=35,000(1.02)^{n} $$ (a) Find the first five terms of the sequence. (b) Find the population in 2014 .

A couple needs a mortgage of \(\$ 300,000\). Their mortgage broker presents them with two options: a 30-year mortgage at \(6 \frac{1}{2} \%\) interest or a 15-year mortgage at \(5 \frac{3}{4} \%\) interest. (a) Find the monthly payment on the 30 -year mortgage and on the 15 -year mortgage. Which mortgage has the larger monthly payment? (b) Find the total amount to be paid over the life of each loan. Which mortgage has the lower total payment over its lifetime?
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