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Chapter 12: Problem 13

Find a general term for each geometric sequence. $$ 10,-2, \frac{2}{5}, \ldots $$

Short Answer

Expert verified
The general term is \(a_n = 10 \times \big(-\frac{1}{5}\big)^{n-1}\).

Step by step solution

01

Identify the first term

The first term of the sequence, denoted as \(a_1\), is given by the first number in the sequence. Here, the first term is \(a_1 = 10\).
02

Determine the common ratio

The common ratio, denoted as \(r\), can be found by dividing the second term by the first term. Here, \(r = \frac{a_2}{a_1} = \frac{-2}{10} = -\frac{1}{5}\).
03

Write the general term formula for a geometric sequence

The general term \(a_n\) of a geometric sequence is given by the formula: \(a_n = a_1 \times r^{n-1}\).
04

Substitute the values into the general term formula

Using \(a_1 = 10\) and \(r = -\frac{1}{5}\), the general term is: \(a_n = 10 \times \big(-\frac{1}{5}\big)^{n-1}\).

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Key Concepts

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

geometric sequences
A geometric sequence is a list 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. For example, in the sequence 10, -2, \(\frac{2}{5}\), ... each term is obtained by multiplying the previous term by -\(\frac{1}{5}\). Geometric sequences are useful in various real-world contexts, such as calculating interest, population growth, or anything that involves exponential growth or decay.
common ratio
The common ratio in a geometric sequence is the factor that you multiply by to get from one term to the next. It's a constant value, and it can be found by dividing any term by the preceding one. In our example, to find the common ratio, you can divide the second term by the first term:
\( r = \frac{a_2}{a_1} \). Here, \( r = \frac{-2}{10} = -\frac{1}{5} \). This means that to get from 10 to -2, we multiply by \(-\frac{1}{5}\), and similarly, from -2 to \(\frac{2}{5}\), we multiply by \(-\frac{1}{5}\). Understanding the common ratio is essential because it helps in identifying the pattern of the sequence and predicting future terms.
general term formula
In a geometric sequence, the general term formula allows you to find any term in the sequence without listing all the terms. The formula is:
\(a_n = a_1 \times r^{n-1} \).
Here's what each part means:
  • \(a_n\) is the nth term of the sequence.
  • \(a_1\) is the first term of the sequence.
  • \(r\) is the common ratio.
  • \(n-1\) is the exponent where n is the term number.
For our example, the first term (\(a_1\)) is 10 and the common ratio (\(r\)) is \(-\frac{1}{5}\). Plugging these into our formula, we get
\(a_n = 10 \times \big(-\frac{1}{5}\big)^{n-1} \). This formula lets us compute any term in the sequence. For instance, the 4th term would be \(a_4 = 10 \times \big(-\frac{1}{5}\big)^{3} = 10 \times -\frac{1}{125} = -\frac{2}{25}\).

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Most popular questions from this chapter

Find a general term \(a_{n}\) for the given terms of each sequence. $$ \frac{2}{5}, \frac{3}{6}, \frac{4}{7}, \frac{5}{8}, \ldots $$

Evaluate each expression. $$ _{13} C_{2} $$

Find the arithmetic mean for each collection of numbers. $$ 8,11,14,9,7,6,8 $$

Write out each series and evaluate it. $$ \sum_{i=1}^{6}(i+9) $$

Find the indicated term for each sequence. $$ a_{n}=(5 n-2)(3 n+1) ; \quad a_{10} $$
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