Use the recursively defined geometric sequence a1=56

, an=4an−1
and find the common ratio.(1 point)
Responses

4
4

−23
negative Start Fraction 2 over 3 End Fraction

56
Start Fraction 5 over 6 End Fraction

103
Start Fraction 10 over 3 End Fraction

1 answer

In a geometric sequence, the common ratio \( r \) can be found from the recursive definition. Given the definition:

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

The common ratio \( r \) is the factor by which we multiply the previous term to get the next term.

From the definition \( a_n = 4a_{n-1} \), we can see that:

  • The second term \( a_2 = 4a_1 = 4 \times 56 = 224 \)
  • The common ratio \( r = \frac{a_n}{a_{n-1}} = \frac{4a_{n-1}}{a_{n-1}} = 4 \)

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

Therefore, the answer is:

4