Question

How do you know if the sequence `1, -2, 4, -8, ...` is arithmetic or geometric?

Answer 1

Because the sequence has a common ratio `r=-2`, it is geometric.

Explanation:

Each term of an arithmetic sequence is generated by adding or subtracting a number to get the next term. The number is called the common difference `d`.

Each term of a geometric sequence is generated by multiplying or dividing by a number to get the next term. The number is called the common ratio `r`.

Let's look at the given sequence: `1, -2, 4, -8...`

If you subtract the first term from the second, `-2-1=3`. and the second term from the third, `4- -2= 6`, you can see that there is no common difference between the terms. The difference between these terms is not the same. The sequence is not arithmetic.

If you divide the second term by the first, `(-2)/1=-2`, and the third term by the second, `4/-2=-2`, you get the same quotient. In fact, if you divide the fourth term by the third, `(-8)/4=-2`, the pattern continues. The common ratio is `r=-2`.

Because there is a common ratio, the sequence is geometric