Question

3. Use the recursively defined geometric sequence a1 = 5/6, an = 4an-1 and find the common ratio.
A) 10/3
B) 4
C) 5/6
D) -2/3

Answer 1

To find the common ratio of the given recursively defined geometric sequence, we start with the first term provided, \( a_1 = \frac{5}{6} \), and the recursive definition \( a_n = 4a_{n-1} \).

In a geometric sequence, the common ratio \( r \) can be determined by the relationship of consecutive terms, i.e., \( r = \frac{a_n}{a_{n-1}} \).

Using the recursive definition:
\[
a_2 = 4a_1
\]
We calculate \( a_2 \):
\[
a_2 = 4 \cdot \frac{5}{6} = \frac{20}{6} = \frac{10}{3}
\]

Now we can calculate the common ratio \( r \):
\[
r = \frac{a_2}{a_1} = \frac{\frac{10}{3}}{\frac{5}{6}} = \frac{10}{3} \cdot \frac{6}{5} = \frac{10 \cdot 6}{3 \cdot 5} = \frac{60}{15} = 4
\]

The calculation shows that:
\[
r = 4
\]

Thus, the common ratio of the given geometric sequence is \(\boxed{4}\).