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Chapter 14: Problem 40

Find the indicated term of each sequence. The fifth term of the geometric sequence whose first term is 8 and whose common ratio is \(-3\)

Short Answer

Expert verified
The fifth term is 648.

Step by step solution

01

Understand the Formula for the nth Term

The nth term of a geometric sequence can be found using the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
02

Identify Given Values

Here, we have been given \( a_1 = 8 \), \( r = -3 \), and \( n = 5 \). We need to find \( a_5 \), the fifth term of the sequence.
03

Plug Values into the Formula

Substitute the given values into the formula: \[ a_5 = 8 \cdot (-3)^{(5-1)} \].
04

Simplify the Exponent

Calculate the exponent: \((-3)^4 = 81\).
05

Calculate the Fifth Term

Now compute \( a_5 = 8 \cdot 81 = 648 \).
06

Determine the Sign

Since \((-3)^4\) is even, the result is positive, confirming that \( a_5 = 648 \).

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Key Concepts

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

nth term formula
The nth term formula for a geometric sequence is a powerful tool that helps you find any term in the sequence without listing all terms. It is given by the expression \( a_n = a_1 \cdot r^{(n-1)} \), where:
  • \( a_n \) is the term you want to find.
  • \( a_1 \) is the first term of the sequence.
  • \( r \) is the common ratio.
  • \( n \) is the position of the term within the sequence.
By plugging the appropriate values into this formula, you can quickly determine any term in the sequence. This saves time and effort, especially in large sequences. Remember, understanding the formula and its components is crucial to getting correct results.
common ratio
The common ratio \( r \) in a geometric sequence is the factor by which each term is multiplied to get the next term. It is a key characteristic that defines the behavior of the sequence. Here's how it works:
  • To find the common ratio, divide any term by its preceding term.
  • If the sequence is geometric, this ratio will remain constant throughout.
  • In our exercise, the common ratio is given as \(-3\), which means each term is multiplied by \(-3\) to get the next term.
Knowing the common ratio allows you to predict how the sequence will progress. With this knowledge, you can easily calculate any term using the formula or recognize patterns in the sequence.
exponentiation
Exponentiation is a mathematical operation involving numbers being raised to a power. In the context of geometric sequences, exponentiation is used to express repeated multiplication of the common ratio. For instance, if you raise the common ratio to the power of \( n-1 \), you are calculating the factor that transforms the first term into the nth term.
  • The notation \( r^{(n-1)} \) represents repeated multiplication of \( r \) \( n-1 \) times.
  • In our example, since \( r = -3 \) and \( n = 5 \), we compute \((-3)^4\).
  • Results of exponentiation can significantly change based on whether the exponent is even or odd, since it affects the sign of the term.
Understanding exponentiation is crucial for calculating terms correctly and ensures you grasp how the sequence behaves over different positions.
term calculation
Term calculation in a geometric sequence involves using both the nth term formula and exponentiation. By substituting the known values into the formula and solving step-by-step, each term of the sequence can be calculated effectively.
  • First, substitute the values into the formula: \( a_5 = 8 \cdot (-3)^{(5-1)} \).
  • Next, calculate the power: \( (-3)^4 = 81 \).
  • Finally, multiply by the first term: \( 8 \cdot 81 = 648 \).
Double-check the sign of your answer, especially when working with negative ratios and even exponents, to ensure accuracy. This understanding not only aids in performing calculations accurately but also highlights the mechanics of how each term is derived from its predecessors.

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