To solve this problem, we need to find the [tex]\(12^\text{th}\)[/tex] term of a geometric sequence (GS) where the [tex]\(13^\text{th}\)[/tex] term is given as 625 and the common ratio is 5.
In a geometric sequence, the [tex]\(n^\text{th}\)[/tex] term can be found using the formula:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n^\text{th}\)[/tex] term.
- [tex]\(a\)[/tex] is the first term.
- [tex]\(r\)[/tex] is the common ratio.
- [tex]\(n\)[/tex] is the term number.
Given:
- [tex]\(a_{13} = 625\)[/tex]
- [tex]\(r = 5\)[/tex]
First, we need to find the first term [tex]\(a\)[/tex] of the geometric sequence.
We know:
[tex]\[ a_{13} = a \cdot r^{12} \][/tex]
Substitute the given values:
[tex]\[ 625 = a \cdot 5^{12} \][/tex]
To find [tex]\(a\)[/tex], we solve for it by dividing both sides of the equation by [tex]\(5^{12}\)[/tex]:
[tex]\[ a = \frac{625}{5^{12}} \][/tex]
Now that we have the first term [tex]\(a\)[/tex], we can use it to find the [tex]\(12^\text{th}\)[/tex] term [tex]\(a_{12}\)[/tex].
Using the formula:
[tex]\[ a_{12} = a \cdot r^{11} \][/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(r\)[/tex] into the formula:
[tex]\[ a_{12} = \left(\frac{625}{5^{12}}\right) \cdot 5^{11} \][/tex]
Simplify the expression:
[tex]\[ a_{12} = \frac{625 \cdot 5^{11}}{5^{12}} \][/tex]
[tex]\[ a_{12} = \frac{625}{5} \][/tex]
[tex]\[ a_{12} = 125 \][/tex]
Therefore, the [tex]\(12^\text{th}\)[/tex] term of the geometric sequence is [tex]\(125\)[/tex].
