Question

Determine if the sequence is a geometric sequence. If it is, find the common ratio and write the explicit formula and recursive definition.

7
112, 28, 7.
Determine if the sequence is a geometric sequence. If it is, find the common ratio. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The sequence is a geometric sequence. The common ratio, r=
(Type an integer or a fraction.)
B. The sequence is not a geometric sequence.

Answer 1

The sequence is a geometric sequence with a common ratio of 4. The explicit formula is an = 112 × 4^(n-1), and the recursive definition is an = 4 × a(n-1) for n > 1.

Step-by-step explanation:

To determine if the sequence is geometric, we need to check if there is a constant ratio between successive terms. We divide each term by the previous one to find this common ratio:

• 112 ÷ 28 = 4
• 28 ÷ 7 = 4

Since each term is obtained by multiplying the previous term by the same number, we confirm that this is indeed a geometric sequence with a common ratio of 4.

The explicit formula for the nth term of a geometric sequence is given by:

an = a1 × r(n-1)

For this sequence, a1 = 112 and r = ⅒. The formula becomes:

an = 112 × 4(n-1)

The recursive definition would be:

a1 = 112

an = 4 × a(n-1), for n > 1