Certainly! Let's examine the given sequences in detail:
1. First Sequence
- First Term: [tex]\( a_1 = 3 \)[/tex]
- Common Ratio: [tex]\( r_1 = 2 \)[/tex]
A geometric sequence follows the pattern: [tex]\( a, ar, ar^2, ar^3, \ldots \)[/tex]
Applying this to the first sequence:
- First Term: [tex]\( 3 \)[/tex]
- Second Term: [tex]\( 3 \times 2 = 6 \)[/tex]
- Third Term: [tex]\( 3 \times 2^2 = 3 \times 4 = 12 \)[/tex]
- and so on.
2. Second Sequence
- First Term: [tex]\( a_2 = -2 \)[/tex]
- Common Ratio: [tex]\( r_2 = \frac{1}{4} \)[/tex]
Similarly, applying this to the second sequence:
- First Term: [tex]\( -2 \)[/tex]
- Second Term: [tex]\( -2 \times \frac{1}{4} = -\frac{2}{4} = -0.5 \)[/tex]
- Third Term: [tex]\( -2 \times \left(\frac{1}{4}\right)^2 = -2 \times \frac{1}{16} = -\frac{2}{16} = -\frac{1}{8} \)[/tex]
- and so on.
### Let's break down the results provided in the prompt:
- For the first sequence:
- The first term [tex]\(a_1\)[/tex] is 3.
- The common ratio [tex]\(r_1\)[/tex] is 2.
- The second term of the sequence can be found by multiplying the first term by the common ratio: [tex]\( 3 \times 2 = 6 \)[/tex].
- For the second sequence:
- The first term [tex]\(a_2\)[/tex] is -2.
- The common ratio [tex]\(r_2\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
- The second term of the sequence can be found by multiplying the first term by the common ratio: [tex]\( -2 \times \frac{1}{4} = -0.5 \)[/tex].
### Summary of Results:
- For the first sequence:
- First term: 3
- Common ratio: 2
- Example second term: 6
- For the second sequence:
- First term: -2
- Common ratio: [tex]\(\frac{1}{4}\)[/tex]
- Example second term: -0.5
These geometric sequences demonstrate the typical behavior of exponential growth and decay respectively, based on their common ratios.
