If the 12th term of a sequence is 144 and the 8th term is 64, is the sequence arithmetic, geometric, or neither?*GeometricNeitherArithmetic

To determine whether the sequence is arithmetic, geometric, or neither, we need to understand the properties of each type of sequence.

1. Arithmetic Sequence: In an arithmetic sequence, the difference between any two consecutive terms is constant. This is called the common difference.

2. Geometric Sequence: In a geometric sequence, the ratio of any two consecutive terms is constant. This is called the common ratio.

Now, let's check the given sequence.

The 12th term is 144 and the 8th term is 64.

If the sequence was arithmetic, the difference between consecutive terms would be constant. We can calculate this difference by subtracting the 8th term from the 12th term and dividing by the number of terms between them:

(144 - 64) / (12 - 8) = 80 / 4 = 20

However, this would mean that the 9th term would be 64 + 20 = 84, the 10th term would be 84 + 20 = 104, the 11th term would be 104 + 20 = 124, and the 12th term would be 124 + 20 = 144. This is not the case, so the sequence is not arithmetic.

If the sequence was geometric, the ratio between consecutive terms would be constant. We can calculate this ratio by dividing the 12th term by the 8th term and taking the fourth root (since there are four terms between the 8th and 12th terms):

(144 / 64)^(1/4) = 1.5

However, this would mean that the 9th term would be 64 * 1.5 = 96, the 10th term would be 96 * 1.5 = 144, the 11th term would be 144 * 1.5 = 216, and the 12th term would be 216 * 1.5 = 324. This is not the case, so the sequence is not geometric.

Therefore, the sequence is neither arithmetic nor geometric.