To find the 5th term of the given geometric sequence [tex]\(3, \frac{3}{4}, \frac{3}{16}, \ldots\)[/tex]:
1. Identify the first term: The first term of the sequence ([tex]\(a_1\)[/tex]) is 3.
2. Determine the common ratio: The common ratio ([tex]\(r\)[/tex]) can be found by dividing the second term by the first term.
[tex]\[
r = \frac{\frac{3}{4}}{3} = \frac{1}{4}
\][/tex]
3. Use the formula for the nth term of a geometric sequence: The nth term of a geometric sequence is given by [tex]\(a_n = a_1 \cdot r^{(n-1)}\)[/tex].
Let's find the 5th term ([tex]\(a_5\)[/tex]):
[tex]\[
a_5 = a_1 \cdot r^{(5-1)} = 3 \cdot \left(\frac{1}{4}\right)^4
\][/tex]
4. Calculate the power:
[tex]\[
\left(\frac{1}{4}\right)^4 = \frac{1^4}{4^4} = \frac{1}{256}
\][/tex]
5. Multiply to find the 5th term:
[tex]\[
a_5 = 3 \cdot \frac{1}{256} = \frac{3}{256}
\][/tex]
So, the 5th term ([tex]\(a_5\)[/tex]) is [tex]\( \boxed{\frac{3}{256}} \)[/tex].
