Question

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).
36, comma, 27, comma, start fraction, 81, divided by, 4, end fraction, comma, point, point, .
36,27,
4
81
​
,...
Find the 10th term.
Find the 10th term.

Answer 1

To find the 10th term of the sequence given the first three terms: \( 36, 27, \frac{81}{4} \), we first need to identify the pattern in the sequence.

1. The first term is \( a_1 = 36 \).
2. The second term is \( a_2 = 27 \).
3. The third term is \( a_3 = \frac{81}{4} = 20.25 \).

Next, we will examine the sequence to see if there is a consistent pattern or formula. Let's look at the ratios between the terms:

- The ratio between \( a_1 \) and \( a_2 \) is:
\[
\frac{a_2}{a_1} = \frac{27}{36} = \frac{3}{4}
\]

- The ratio between \( a_2 \) and \( a_3 \) is:
\[
\frac{a_3}{a_2} = \frac{\frac{81}{4}}{27} = \frac{81}{4 \times 27} = \frac{81}{108} = \frac{3}{4}
\]

We see that each term is multiplied by \( \frac{3}{4} \) to get to the next term.

Using this ratio, we can express the \( n \)-th term of the sequence:
\[
a_n = a_1 \left( \frac{3}{4} \right)^{n-1} = 36 \left( \frac{3}{4} \right)^{n-1}
\]

To find the 10th term \( a_{10} \):
\[
a_{10} = 36 \left( \frac{3}{4} \right)^{9}
\]

Now we calculate \( \left( \frac{3}{4} \right)^{9} \):
\[
\left( \frac{3}{4} \right)^{9} = \frac{3^9}{4^9} = \frac{19683}{262144}
\]

Next, we multiply by 36:
\[
a_{10} = 36 \times \frac{19683}{262144} = \frac{36 \times 19683}{262144} = \frac{708708}{262144}
\]

Now we perform the division:
\[
708708 \div 262144 \approx 2.703
\]

Finally, rounding to the nearest thousandth:
\[
\boxed{2.703}
\]

So, the 10th term of the sequence is approximately \( 2.703 \).