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Chapter 12: Problem 189

In your own words, explain how to find the general term of a geometric sequence.

Short Answer

Expert verified
Use \( a_n = a_1 \times r^{(n-1)} \) where a_1 is the first term and r is the common ratio.

Step by step solution

01

- Understand a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
02

- Identify Key Elements

Identify the first term of the sequence, denoted as a_1, and the common ratio, denoted as r.
03

- Write the General Term Formula

The general term of a geometric sequence can be written as: \[ a_n = a_1 \times r^{(n-1)} \] where a_n represents the n-th term of the sequence.
04

- Plug in Values

To find a specific term, substitute the values of a_1, r, and n into the general term formula. For example, to find the 5th term (a_5), use the formula with n = 5.

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Key Concepts

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

General Term Formula
To fully understand how to find the general term of a geometric sequence, we must first understand what the general term formula represents. The general term formula for a geometric sequence is used to find any term in the sequence without having to go through all the previous terms.

Here鈥檚 the formula: \[ a_n = a_1 \times r^{(n-1)} \]

In this formula:
  • an represents the n-th term you want to find.
  • a1 is the first term of the sequence.
  • r is the common ratio.
  • n is the term number.
Using this formula, you can quickly find the position of any term in the sequence without needing to list all previous terms. For example, to find the 5th term, you use n = 5.
Common Ratio
The common ratio is a crucial part of a geometric sequence. It is the fixed, non-zero number that you multiply each term by to get the next term.

To identify the common ratio r in a sequence:
  • Choose any term in the sequence except the first one.
  • Divide that term by the one that comes immediately before it.
For example, if you have the sequence 2, 6, 18, 54, you can find r by dividing 6 by 2 or 18 by 6, which both give you 3. Hence, the common ratio r is 3.

Having the correct common ratio is vital for correctly calculating any term in the sequence using the general term formula.
Sequence Terms
Taking the sequence terms from the general term formula and the common ratio, you can construct a sequential list of numbers in a geometric sequence. For example, if the first term a1 is 2 and the common ratio r is 3, the first few terms would be:

  • First term (a1): 2
  • Second term (a2): 2 x 3 = 6
  • Third term (a3): 6 x 3 = 18
  • Fourth term (a4): 18 x 3 = 54
Notice how each term can be easily calculated using the term before it and multiplying by the common ratio.

If you need to find the 10th term directly, you can use the general term formula instead of calculating each step sequentially. Plugging in a1, r, and n = 10 into the formula gives:
\[ a_{10} = 2 \times 3^{(10-1)} = 2 \times 3^9 \].

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