Question

If the following is a geometric sequence, find r and aₙ​. −2/9,2/15​,−2/25​,6/125​,… Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. r= (Simplify your answer. Type an integer or a simplified fraction.) B. The sequence is not a geometric sequence.

Answer 1

The sequence is geometric with a common ratio of r = −3/5. To find the nth term (an) of the sequence, use the formula an = (−2/9) × (−3/5)n-1, replacing n with the desired term number.

Step-by-step explanation:

To determine whether a sequence is geometric, we must identify a common ratio (r) that when multiplied to a term gives us the next term in the sequence. Looking at the provided sequence −2/9, 2/15, −2/25, 6/125,..., to find the common ratio, we divide the second term by the first term:

r = (2/15) / (−2/9) = −15/2 × 9/2 = −10/3 × 3/1 = −3/5

We can then check this ratio by dividing subsequent terms:

(−2/25) / (2/15) = −15/2 × 15/2 = −3/5

(6/125) / (−2/25) = 25/2 × 2/6 = −3/5

As the ratio is consistent, the sequence is geometric with r = −3/5.

For finding the nth term of the sequence (an), we use the formula an = a1 × rn-1. With a1 = −2/9 and r = −3/5, we have:

an = (−2/9) × (−3/5)n-1

Therefore, to find any term in the sequence, replace n with the desired term number and compute the value.