Question

Use the recursively defined geometric sequence a1=5/6 , an=4an−1 and find the common ratio.(1 point)
Responses

56
Start Fraction 5 over 6 End Fraction

103
Start Fraction 10 over 3 End Fraction

−23
negative Start Fraction 2 over 3 End Fraction

4
4

Answer 1

To find the common ratio of the geometric sequence defined by \( a_1 = \frac{5}{6} \) and \( a_n = 4a_{n-1} \), we start with the recursive formula:

1. **Identify the first term**:
\[
a_1 = \frac{5}{6}
\]

2. **Use the recursive formula to find the next term, \( a_2 \)**:
\[
a_2 = 4a_1 = 4 \left(\frac{5}{6}\right) = \frac{20}{6} = \frac{10}{3}
\]

3. **Now, consider the ratio between consecutive terms**:
The common ratio \( r \) in a geometric sequence is defined as:
\[
r = \frac{a_n}{a_{n-1}}
\]
Specifically for this sequence:
\[
r = \frac{a_2}{a_1} = \frac{\frac{10}{3}}{\frac{5}{6}} = \frac{10}{3} \times \frac{6}{5} = \frac{60}{15} = 4
\]

4. **Conclusion**:
The common ratio \( r \) for the given geometric sequence is \( 4 \).

Therefore, the answer is:
\[
\boxed{4}
\]