Question

) Which of the following correctly uses the explicit formula, \(A_n=\frac{1}{4}+\left(n1\right)\cdot\frac{1}{2}\), to find the 9th term of the described arithmetic sequence? (1 point)
\(4\frac{3}{4}\)
\(8\frac{3}{4}\)
\(8\frac{1}{4}\)
\(4\frac{1}{4}\)

Answer 1

To find the 9th term \(A_9\) of the arithmetic sequence using the explicit formula given by:

\[
A_n = \frac{1}{4} + (n-1) \cdot \frac{1}{2}
\]

we will substitute \(n = 9\) into the formula.

1. Substitute \(n = 9\):

\[
A_9 = \frac{1}{4} + (9 - 1) \cdot \frac{1}{2}
\]

2. Simplify:

\[
A_9 = \frac{1}{4} + 8 \cdot \frac{1}{2}
\]

3. Calculate \(8 \cdot \frac{1}{2}\):

\[
8 \cdot \frac{1}{2} = 4
\]

4. Now substitute back into the equation:

\[
A_9 = \frac{1}{4} + 4
\]

5. Convert \(4\) to a fraction:

\[
4 = \frac{16}{4}
\]

6. Now add the fractions:

\[
A_9 = \frac{1}{4} + \frac{16}{4} = \frac{17}{4}
\]

7. Convert \(\frac{17}{4}\) to a mixed number:

\[
\frac{17}{4} = 4\frac{1}{4}
\]

Thus, the 9th term \(A_9\) is \(4\frac{1}{4}\).

The correct answer is:
\(4\frac{1}{4}\)