Sure, let's find the 13th term of the geometric sequence [tex]\(3, -15, 75, \ldots\)[/tex].
To solve this, we need to identify the first term and the common ratio of the sequence, and then use the formula for the [tex]\(n\)[/tex]th term of a geometric sequence.
1. Identify the first term:
Here, the first term [tex]\(a\)[/tex] is given as 3.
2. Determine the common ratio [tex]\(r\)[/tex]:
The common ratio can be found by dividing the second term by the first term or the third term by the second term.
[tex]\[
r = \frac{-15}{3} = -5
\][/tex]
3. Apply the formula for the [tex]\(n\)[/tex]th term of a geometric sequence:
The formula for the [tex]\(n\)[/tex]th term [tex]\(a_n\)[/tex] of a geometric sequence is:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
where [tex]\(a\)[/tex] is the first term and [tex]\(r\)[/tex] is the common ratio.
For the 13th term ([tex]\(n = 13\)[/tex]):
[tex]\[
a_{13} = 3 \cdot (-5)^{12}
\][/tex]
4. Simplify the exponentiation and multiplication:
Notice that [tex]\((-5)^{12}\)[/tex] is a large number. Simplifying directly:
[tex]\[
(-5)^{12} = 244140625
\][/tex]
Therefore:
[tex]\[
a_{13} = 3 \cdot 244140625 = 732421875
\][/tex]
So, the 13th term of the geometric sequence [tex]\(3, -15, 75, \ldots\)[/tex] is:
[tex]\[
\boxed{732421875}
\][/tex]
