To find the [tex]\(10^{\text{th}}\)[/tex] term of the geometric sequence [tex]\(3, 15, 75, 375, \ldots\)[/tex], let's break down the process step-by-step.
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term [tex]\(a_1\)[/tex] is clearly given as [tex]\(3\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
The common ratio [tex]\(r\)[/tex] is found by dividing the second term by the first term.
[tex]\[
r = \frac{15}{3} = 5
\][/tex]
3. Use the geometric sequence formula to find the [tex]\(n^{\text{th}}\)[/tex] term:
The general formula to find the [tex]\(n^{\text{th}}\)[/tex] term ([tex]\(a_n\)[/tex]) of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Here, we need to find the [tex]\(10^{\text{th}}\)[/tex] term ([tex]\(a_{10}\)[/tex]). We know [tex]\(a_1 = 3\)[/tex], [tex]\(r = 5\)[/tex], and [tex]\(n = 10\)[/tex].
4. Substitute the values into the formula:
[tex]\[
a_{10} = 3 \cdot 5^{(10-1)}
\][/tex]
Simplify the exponent:
[tex]\[
a_{10} = 3 \cdot 5^9
\][/tex]
Calculate [tex]\(5^9\)[/tex]:
[tex]\[
5^9 = 1953125
\][/tex]
5. Multiply by the first term:
[tex]\[
a_{10} = 3 \cdot 1953125 = 5859375
\][/tex]
Thus, the [tex]\(10^{\text{th}}\)[/tex] term of the geometric sequence is [tex]\(5859375\)[/tex].
