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Chapter 13: Problem 32

Find the indicated term of each geometric sequence. 7th term of \(0.1,1.0,10.0, \ldots\)

Short Answer

Expert verified
The 7th term is 100,000.

Step by step solution

01

Identify the first term and common ratio

The first term (\r a_1\r ) of the sequence is 0.1. To find the common ratio (\r r\r ), divide the second term by the first term: \( r = \frac{1.0}{0.1} = 10 \).
02

Use the geometric sequence formula

The formula to find the nth term in a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \). In this case, we need to find the 7th term.
03

Substitute the values into the formula

Using the values for the 7th term: \( a_7 = 0.1 \cdot 10^{7-1} \).
04

Calculate the exponent

Calculate \( 10^{6} \): \( 10^{6} = 1,000,000 \).
05

Multiply the results

Finally, multiply 0.1 by 1,000,000: \( 0.1 \cdot 1,000,000 = 100,000 \).

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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, the common ratio plays a crucial role. It is the factor by which we multiply each term to get to the next term. To find the common ratio, simply divide any term by the preceding term.
For instance, in the sequence 0.1, 1.0, 10.0, ..., to find the common ratio, divide 1.0 by 0.1. This gives us:
\( r = \frac{1.0}{0.1} = 10 \).
Knowing the common ratio helps in easily determining the progression of the sequence.
geometric sequence formula
The geometric sequence formula is your main tool for finding any term in a geometric sequence. The general form is:
\[ a_n = a_1 \cdot r^{n-1} \].
Here's what each term represents:
  • \(a_n\) is the term you want to find.

  • \(a_1\) is the first term of the sequence.

  • \(r\) is the common ratio.

  • \(n-1\) indicates that the exponent is one less than the term's position.
By substituting the known values into this formula, you can find any term in the sequence.
nth term computation
Computing the nth term requires substituting the values you know into the geometric sequence formula. Let's take the example of finding the 7th term in the given sequence 0.1, 1.0, 10.0, ...
First, identify the values:
  • \(a_1 = 0.1\)

  • \(r = 10\)

  • We need the 7th term, so \(n = 7\)
Using the formula:
\[ a_7 = 0.1 \cdot 10^{7-1} \].
Next, compute the exponent.
exponentiation
Exponentiation is the process of raising a number to a power. In our example, we need to compute \(10^{6}\).
Exponentiation works as repeated multiplication. So, \(10^{6} = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000\).
Now, multiply this result by the first term \(0.1\):
\ 0.1 \cdot 1,000,000 = 100,000 \.
Thus, the 7th term in the sequence is 100,000. Breaking down each step makes it easier to understand the process of finding any term in a geometric sequence.

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Most popular questions from this chapter

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(e^{3 x-7}=4\)

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