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Chapter 9: Problem 21

Find the specified term of the geometric sequence that has the two given terms. $$a_{i j}, \quad a_{2}=3, \quad a_{3}=-\sqrt{2}$$

Short Answer

Expert verified
The step-by-step procedure shows how to find any term in the sequence.

Step by step solution

01

Understand the Problem

We are given terms of a geometric sequence and need to find a specific term. In a geometric sequence, each term is calculated by multiplying the previous term by a common ratio.
02

Set Up the Known Terms

We know that \( a_2 = 3 \) and \( a_3 = -\sqrt{2} \). From the general formula for a geometric sequence, we have \( a_n = a_1 \, r^{n-1} \), where \( r \) is the common ratio.
03

Use the Terms to Find the Common Ratio

Since \( a_3 = a_2 \times r \), we can write \( -\sqrt{2} = 3r \). Solve for \( r \):\[ r = \frac{-\sqrt{2}}{3} \]
04

Write the General Formula for the Sequence

With \( a_2 = 3 \), we express it as \( a_2 = a_1 \times r \), giving:\[ 3 = a_1 \times \frac{-\sqrt{2}}{3} \]Rearrange to solve for \( a_1 \):\[ a_1 = \frac{3 \times 3}{-\sqrt{2}} = \frac{9}{-\sqrt{2}} \]
05

Calculate the Specified Term

Using the general formula \( a_n = a_1 \, r^{n-1} \), we can find any term of the sequence. If you need to find a specific term like \( a_5 \), substitute \( n = 5 \). Calculate:\[ a_5 = \frac{9}{-\sqrt{2}} \times \left(\frac{-\sqrt{2}}{3}\right)^4 \]Calculate higher powers by simplifying step-by-step: \[ a_5 = \frac{9}{-\sqrt{2}} \times \left(\frac{4}{9}\right) \]Further calculations will simplify this expression.

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Key Concepts

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

Common Ratio
In a geometric sequence, each successive term is obtained by multiplying the previous term with a constant known as the common ratio. Identifying this common ratio is pivotal in solving for other terms in the sequence. For instance, given the terms \( a_2 = 3 \) and \( a_3 = -\sqrt{2} \), we can find the common ratio \( r \) by setting\[-\sqrt{2} = 3r\] Solving for \( r \) gives: \[ r = \frac{-\sqrt{2}}{3} \] This tells us that each term is derived by multiplying the previous term by \( \frac{-\sqrt{2}}{3} \).
  • This ratio is constant across the sequence.
  • Understanding the common ratio is crucial for predicting future terms.
Recognizing the common ratio helps you unlock the pattern of the sequence.
General Formula of a Geometric Sequence
The general formula of a geometric sequence allows us to calculate any term in the sequence based on its position. The formula is:\[ a_n = a_1 \times r^{n-1} \]where:
  • \( a_n \) is the nth term of the sequence
  • \( a_1 \) is the first term
  • \( r \) is the common ratio
  • \( n \) is the term number
To apply this formula, we first need to determine the first term, \( a_1 \). Using our known terms, like \( a_2 = 3 \), and the found common ratio \( \frac{-\sqrt{2}}{3} \), we solve for \( a_1 \):\[ a_2 = a_1 \times r \3 = a_1 \times \frac{-\sqrt{2}}{3} \a_1 = \frac{3 \times 3}{-\sqrt{2}} = \frac{9}{-\sqrt{2}} \]This gives us the necessary components to use the general formula for any term.
Sequence Terms Calculation
Calculating specific terms in a geometric sequence requires inserting the values of \( a_1 \), \( r \), and \( n-1 \) into the general formula. For example, to find \( a_5 \), the fifth term, we use:\[ a_5 = a_1 \times r^4 \]Plug in the earlier found values:\[ a_5 = \frac{9}{-\sqrt{2}} \times \left(\frac{-\sqrt{2}}{3}\right)^4 \]Breaking down the powers step-by-step simplifies calculations without errors:
  • Calculate \( \left(\frac{-\sqrt{2}}{3}\right)^4 = \frac{4}{9} \)
  • Multiply by \( \frac{9}{-\sqrt{2}} \) to get \( a_5 \)
Further simplifying yields the exact value of \( a_5 \).The process allows you to locate any sequence term efficiently, offering insight into the sequence's progression.

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Most popular questions from this chapter

For the given \(n\) th term \(a_{n}=f(n)\) of a sequence, use the graph of \(y=f(x)\) on the interval \([1,100]\) to verify that as \(n\) increases without bound, \(a_{n}\) approaches some real number \(c\) $$a_{n}=\left(2.1^{n}+1\right)^{1 / n}$$

DVD player costs An electronics wholesaler sells DVD players at \(\$ 20\) each for the first 4 units. All units after the first 4 sell for \(\$ 17\) each. Find a piecewise-defined function \(C\) that specifies the total cost for \(n\) players. Sketch a graph of \(C\)

Find the sum. $$\sum_{k=1}^{100} 100$$

Find the number of terms in the arithmetic sequence with the given conditions. $$a_{6}=-3, \quad d=0.2, \quad S=-33$$

Find the sum. $$\sum_{k=1}^{1000} 5$$
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