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Chapter 14: Problem 48

Find the indicated term of each sequence. If the third term of a geometric sequence is \(-28\) and the fourth term is \(-56,\) find \(a_{1}\) and \(r\)

Short Answer

Expert verified
The first term is \( a_1 = -7 \) and the common ratio is \( r = 2 \).

Step by step solution

01

Define the Terms of the Sequence

In a geometric sequence, each term can be expressed as \( a_n = a_1 imes r^{n-1} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.
02

Write Equations for Given Terms

We know the third term \( a_3 = -28 \) and the fourth term \( a_4 = -56 \). So, we can write:\[ a_3 = a_1 imes r^2 = -28 \]\[ a_4 = a_1 imes r^3 = -56 \]
03

Create an Equation to Find the Ratio (r)

Divide the equation for \( a_4 \) by the equation for \( a_3 \) to eliminate \( a_1 \):\[ \frac{a_1 r^3}{a_1 r^2} = \frac{-56}{-28} \]\[ r = 2 \]
04

Use Ratio to Find First Term (a1)

Substitute \( r = 2 \) back into the equation for \( a_3 \):\[ a_1 imes (2)^2 = -28 \]\[ a_1 imes 4 = -28 \]\[ a_1 = -7 \]
05

Verify the Solution

Check the values by substituting \( a_1 = -7 \) and \( r = 2 \) back into both original expressions:\[ a_3 = -7 imes 2^2 = -28 \]\[ a_4 = -7 imes 2^3 = -56 \]These match the given terms, confirming our solution is correct.

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Key Concepts

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

Terms of a Sequence
In a geometric sequence, each term is defined by its position and relation to the first term and a constant ratio known as the common ratio. Each term is derived from the formula:
  • \( a_n = a_1 \times r^{n-1} \)

  • where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.
This means that once the first term and the common ratio are known, you can find all terms in the sequence.
For example, if we know the third and fourth terms of a sequence, we can use these to express relationships among the terms, helping solve for unknowns like the first term or the common ratio.
Common Ratio
The common ratio \( r \) in a geometric sequence determines how each term changes from one to the next.
It's a multiplier applied to one term to generate the next. In our original problem, the relationship between the terms provided allows us to find \( r \).
  • If the third term is \(-28\) and the fourth term is \(-56\), the common ratio can be determined by dividing the fourth term by the third term:
\[ r = \frac{a_4}{a_3} = \frac{-56}{-28} = 2 \]This value of \( r \) indicates that each term in the sequence is double the previous term.
Understanding the common ratio is crucial because it fundamentally defines the nature of the geometric sequence.
First Term
The first term, \( a_1 \), is the starting point of a geometric sequence.
Once the common ratio has been identified, \( a_1 \) can be easily calculated using any known term.
  • Using our problem, if \( r = 2 \) and the third term is \(-28\), we back substitute into the formula:

  • \[ a_3 = a_1 \times r^2 \]
  • Plug in the known values:

  • \[ -28 = a_1 \times 4 \]

  • Solving for \( a_1 \):

  • \[ a_1 = \frac{-28}{4} = -7 \]
It's important to cross-verify the calculated \( a_1 \) by substituting back into different term equations to confirm its correctness.
Step by Step Solution
Solving a geometric sequence problem involves careful use of the given terms and formulas.
Here's how you can navigate through solving such problems:
  • Start by identifying the known terms and establishing equations representing them using the sequence formula, \( a_n = a_1 \times r^{n-1} \).

  • Use these equations to isolate and find one unknown, typically the common ratio.

  • If you have consecutive terms like in this problem, dividing one term equation by another often simplifies the process and helps eliminate unwanted variables.

  • Substitute back to find the remaining unknown, such as the first term.

  • Always verify your solution by plugging back into the original equations to ensure consistency with the problem data.
This step-by-step approach ensures that all aspects of the sequence are considered and calculated accurately.

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Most popular questions from this chapter

Solve. An auditorium has 54 seats in the first row, 58 seats in the second row, 62 seats in the third row, and so on. Find the general term of this arithmetic sequence and the number of seats in the twentieth row.

Find the indicated term of each sequence. Explain why \(14,10,\) and 6 may be the first three terms of an arithmetic sequence when it appears we are subtracting instead of adding to get the next term.

Describe a situation in everyday life that can be modeled by an infinite geometric series.

Solve. Find the sum of the first five terms of the sequence \(-2,-6\) \(-18, \ldots\)

Solve. Find the sum of the first fifteen terms of the sequence \(-5,-9,-13, \ldots,-61\) where \(-61\) is the fifteenth term.
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