Question
The first three terms of a geometric sequence are as follows. \[ 162,54,18 \] Find the next two terms of this sequence. \[ 162,54,18, \square, \square \]
Ask by Hilton Herrera. in the United States
Jan 30,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The next two terms are 6 and 2. So the sequence is 162, 54, 18, 6, 2.
Solution
To find the next two terms in the geometric sequence \( 162, 54, 18 \), we'll follow these steps:
1. **Determine the Common Ratio (\( r \))**
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio (\( r \)).
\[
r = \frac{\text{Second Term}}{\text{First Term}} = \frac{54}{162} = \frac{1}{3}
\]
2. **Find the Next Two Terms**
- **Fourth Term (\( a_4 \))**
\[
a_4 = a_3 \times r = 18 \times \frac{1}{3} = 6
\]
- **Fifth Term (\( a_5 \))**
\[
a_5 = a_4 \times r = 6 \times \frac{1}{3} = 2
\]
3. **Complete Sequence**
Adding these terms to the original sequence:
\[
162, \ 54, \ 18, \ 6, \ 2
\]
**Answer:** The next two terms are 6 and 2. So the sequence continues as 162, 54, 18, 6, 2.
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Simplify this solution The Deep Dive
To find the next two terms of the geometric sequence, we first need to determine the common ratio. We can do this by dividing the second term by the first term: \[ r = \frac{54}{162} = \frac{1}{3} \] Now we confirm the ratio with the next pair: \[ r = \frac{18}{54} = \frac{1}{3} \] Since the common ratio \( r \) is \( \frac{1}{3} \), we can find the next two terms by multiplying the last known term (18) by \( \frac{1}{3} \). Calculating the fourth term: \[ 18 \times \frac{1}{3} = 6 \] Now for the fifth term: \[ 6 \times \frac{1}{3} = 2 \] Thus, the next two terms are: \[ 162, 54, 18, 6, 2 \]
