To find the common ratio and the first four terms of the geometric sequence given by [tex]\(\frac{4^{n-3}}{5}\)[/tex], follow these steps:
1. Identify the general term of the sequence:
The general term of the sequence is [tex]\( a_n = \frac{4^{n-3}}{5} \)[/tex].
2. Calculate the common ratio:
The common ratio in a geometric sequence can be found by dividing the second term by the first term.
Let's calculate the values for the first two terms:
- First term ([tex]\(a_1\)[/tex]) for [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = \frac{4^{1-3}}{5} = \frac{4^{-2}}{5} = \frac{1}{16} \cdot \frac{1}{5} = \frac{1}{80} = 0.0125
\][/tex]
- Second term ([tex]\(a_2\)[/tex]) for [tex]\(n = 2\)[/tex]:
[tex]\[
a_2 = \frac{4^{2-3}}{5} = \frac{4^{-1}}{5} = \frac{1}{4} \cdot \frac{1}{5} = \frac{1}{20} = 0.05
\][/tex]
Therefore, the common ratio [tex]\(r\)[/tex] is:
[tex]\[
r = \frac{a_2}{a_1} = \frac{0.05}{0.0125} = 4.0
\][/tex]
So, the common ratio is [tex]\(4.0\)[/tex].
3. Calculate the first four terms:
- First term ([tex]\(a_1\)[/tex]):
[tex]\[
a_1 = \frac{4^{1-3}}{5} = 0.0125
\][/tex]
- Second term ([tex]\(a_2\)[/tex]):
[tex]\[
a_2 = \frac{4^{2-3}}{5} = 0.05
\][/tex]
- Third term ([tex]\(a_3\)[/tex]) for [tex]\(n = 3\)[/tex]:
[tex]\[
a_3 = \frac{4^{3-3}}{5} = \frac{4^{0}}{5} = \frac{1}{5} = 0.2
\][/tex]
- Fourth term ([tex]\(a_4\)[/tex]) for [tex]\(n = 4\)[/tex]:
[tex]\[
a_4 = \frac{4^{4-3}}{5} = \frac{4^{1}}{5} = \frac{4}{5} = 0.8
\][/tex]
So, the values are:
- Common ratio is [tex]\(4.0\)[/tex]
- [tex]\(a_1 = 0.0125\)[/tex]
- [tex]\(a_2 = 0.05\)[/tex]
- [tex]\(a_3 = 0.2\)[/tex]
- [tex]\(a_4 = 0.8\)[/tex]
