To find the 6th term of the given geometric sequence, let's first identify the key parameters of the sequence.
The sequence is:
[tex]\[ 3, \frac{3}{4}, \frac{3}{16}, \ldots \][/tex]
We start with the initial term \( a_1 \):
[tex]\[ a_1 = 3 \][/tex]
Next, we determine the common ratio \( r \) of the sequence. The common ratio can be found by dividing the second term by the first term:
[tex]\[ r = \frac{\frac{3}{4}}{3} = \frac{3}{4} \times \frac{1}{3} = \frac{1}{4} \][/tex]
Now we need to find the 6th term of the sequence, \( a_6 \).
The general formula for the nth term of a geometric sequence is given by:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
For \( n = 6 \):
[tex]\[ a_6 = a_1 \cdot r^{6-1} \][/tex]
[tex]\[ a_6 = 3 \cdot \left(\frac{1}{4}\right)^{5} \][/tex]
Given the common ratio \( r = \frac{1}{4} \), we need to compute \( \left(\frac{1}{4}\right)^5 \):
[tex]\[ \left(\frac{1}{4}\right)^5 = \frac{1}{4^5} = \frac{1}{1024} \][/tex]
Thus, the 6th term \( a_6 \) is:
[tex]\[ a_6 = 3 \cdot \frac{1}{1024} = \frac{3}{1024} \][/tex]
So, the 6th term of the sequence is:
[tex]\[ \boxed{0.0029296875} \][/tex]
