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Chapter 11: Problem 43

Find the indicated term of each sequence. The fifteenth term of the arithmetic sequence \(\frac{3}{2}, 2, \frac{5}{2}, \ldots\)

Short Answer

Expert verified
The 15th term is \(\frac{17}{2}\) or \(8\frac{1}{2}\).

Step by step solution

01

Identify the First Term and the Common Difference

The first term of the sequence is given as \(a_1 = \frac{3}{2}\). The second term is \(2\), so the common difference (\(d\)) can be calculated as follows: \(d = 2 - \frac{3}{2} = \frac{1}{2}\). Thus, the common difference is \(\frac{1}{2}\).
02

Write the General Formula for the n-th Term

The general formula for the n-th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \times d\). Here, \(a_1 = \frac{3}{2}\) and \(d = \frac{1}{2}\).
03

Substitute the Values to Find the 15th Term

Substitute \(n = 15\), \(a_1 = \frac{3}{2}\), and \(d = \frac{1}{2}\) into the formula: \[a_{15} = \frac{3}{2} + (15-1) \times \frac{1}{2}\]
04

Simplify the Expression

First, calculate the multiplication: \((15-1) \times \frac{1}{2} = 14 \times \frac{1}{2} = 7\). Then add to the first term: \[a_{15} = \frac{3}{2} + 7 = \frac{3}{2} + \frac{14}{2} = \frac{17}{2}\]
05

Convert to Mixed Number (Optional)

The fraction \(\frac{17}{2}\) can be expressed as a mixed number: \(\frac{17}{2} = 8\frac{1}{2}\). Thus, the 15th term is \(8\frac{1}{2}\).

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Key Concepts

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

Common Difference
Every arithmetic sequence follows a pattern where the difference between consecutive terms remains constant. This fixed difference is known as the "common difference." For instance, take the sequence given in the exercise:
  • First term: \( \frac{3}{2} \)
  • Second term: 2
To find the common difference \(d\), subtract the first term from the second term:\[ d = 2 - \frac{3}{2} = \frac{1}{2} \]This means each term in the sequence increases by \( \frac{1}{2} \) compared to the previous term. Understanding the common difference is crucial as it helps predict the sequence terms.
N-th Term Formula
To find any term in an arithmetic sequence, use the n-th term formula, a simple mathematical expression:\[ a_n = a_1 + (n-1) \times d \]In this formula:
  • \(a_n\) is the n-th term in the sequence.
  • \(a_1\) is the first term.
  • \(n\) denotes the term position you want to find.
  • \(d\) is the common difference.
So for our exercise, with \(a_1\) as \(\frac{3}{2}\) and \(d\) as \(\frac{1}{2}\), you can substitute these values into the formula for any desired n-th term, simplifying calculations and helping you easily find the intended term.
Sequence Terms
In an arithmetic sequence, each term follows a particular order dictated by its position and the common difference. If you're given a sequence, you can predict future terms by using:
  • The first term (starting point), which is \(\frac{3}{2}\) in our example.
  • The common difference, which is \(\frac{1}{2}\).
For instance, if you're curious about the next few terms after \(\frac{3}{2}\), simply add the common difference repeatedly:
  • Second term: \( \frac{3}{2} + \frac{1}{2} = 2 \)
  • Third term: \( 2 + \frac{1}{2} = \frac{5}{2} \)
  • Fourth term: \( \frac{5}{2} + \frac{1}{2} = 3 \)
Following this pattern, you will be able to find as many sequence terms as needed.
Mixed Numbers
Fractions can sometimes be converted into mixed numbers for easier understanding or simplification. A mixed number consists of a whole number and a proper fraction. For example, in the solution:
  • The 15th term is \( \frac{17}{2} \).
  • The mixed number form is \( 8\frac{1}{2} \).
To convert a fraction like \( \frac{17}{2} \) to a mixed number, divide the numerator (17) by the denominator (2):
  • 17 divided by 2 is 8 with a remainder of 1.
  • This makes the expression \( 8\frac{1}{2} \).
Mixed numbers make interpreting and visualizing quantities easier, especially when dealing with everyday measurements or when simplifying answers is necessary.

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Most popular questions from this chapter

Use the partial sum formula to find the partial sum of the given arithmetic or geometric sequence. See Examples 1 and 4 Find the sum of the first four terms of the arithmetic sequence \(-4,-8,-12, \dots\)

Solve. See Example 5 In free fall, a parachutist falls 16 feet during the first second, 48 feet during the second second, 80 feet during the third second, and so on. Find how far she falls during the eighth second. Find the total distance she falls during the first 8 seconds.

Find partial sum. Find the sum of the first four terms of the sequence whose general term is \(a_{n}=-2 n\).

The number of opossums killed each month on a new highway forms the sequence whose general term is \(a_{n}=(n+1)(n+2),\) where \(n\) is the number of the month. Find the number of opossums killed in the fourth month and find the total number killed in the first four months.

Explain the difference between a sequence and a series.
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