Problem 32 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The 7th term is 100,000.

Step 1: 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: \( r = \frac{1.0}{0.1} = 10 \).

Step 2: 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.

Step 3: Substitute the values into the formula

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

Step 4: Calculate the exponent

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

Step 5: Multiply the results

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

Key Concepts

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.