Use the recursively defined geometric sequence a1=56

, 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

1 answer

In a geometric sequence, the common ratio \( r \) is defined as the factor by which we multiply one term to get the next term in the sequence.

Given the recursively defined sequence:

  • \( a_1 = 56 \)
  • \( a_n = 4a_{n-1} \)

To find the common ratio, let's look at how one term relates to the previous term:

\[ a_2 = 4a_1 = 4 \times 56 = 224 \]

Now, the common ratio \( r \) can be calculated as follows:

\[ r = \frac{a_2}{a_1} = \frac{224}{56} = 4 \]

Thus, the common ratio of the geometric sequence is \( 4 \).

The correct response is:

4