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). 5, comma, start fraction, 5, divided by, 2, end fraction, comma, start fraction, 5, divided by, 4, end fraction, comma, dot dot dot 5, 2 5 ​ , 4 5 ​ ,... Find the 9th term. Find the 9th term.

Answer 1

To find the 9th term of the sequence given as 5, \(\frac{5}{2}\), \(\frac{5}{4}\), ..., we can observe the pattern in the terms.

The sequence starts with:
- 1st term: \(5\)
- 2nd term: \(\frac{5}{2} = 5 \cdot \left(\frac{1}{2}\right)^1\)
- 3rd term: \(\frac{5}{4} = 5 \cdot \left(\frac{1}{2}\right)^2\)

From this, we can surmise that the nth term of the sequence can be expressed as:

\[
a_n = 5 \cdot \left(\frac{1}{2}\right)^{n-1}
\]

Now, we can use this formula to compute the 9th term (\(a_9\)):

\[
a_9 = 5 \cdot \left(\frac{1}{2}\right)^{9-1}
\]
\[
a_9 = 5 \cdot \left(\frac{1}{2}\right)^{8}
\]
\[
a_9 = 5 \cdot \frac{1}{256}
\]
\[
a_9 = \frac{5}{256}
\]

Now, let's calculate this value as a decimal:

\[
\frac{5}{256} \approx 0.01953125
\]

Rounding to the nearest thousandth, we get:

\[
\boxed{0.020}
\]

So the 9th term is approximately \(0.020\).