# At a Glance - Comparing Arithmetic and Geometric Sequences

When we were young, we were all taught addition and subtraction first. Then we endured the endless multiplication tables, and we learned division is multiplication's partner operation.

We did the same with arithmetic and geometric sequences, which differ only in their operations. We went through arithmetic sequences, which are additive. Then we talked about geometric sequences, which are multiplicative. Now we are going to compare them to each other *and* to a tasty treat: a Kit Kat bar.

An arithmetic sequence is one where the *difference* between successive terms is constant.

To determine if a sequence is arithmetic, find the difference between successive terms and see if you always get the same thing. This is like breaking off a piece of a Kit Kat one chocolate-covered wafer at a time.

A geometric sequence is one where the *ratio* between successive terms is constant.

To determine if a sequence is geometric, find the ratio between successive terms and see if you always get the same thing. We want a big Kit Kat bar for this one. If we break one piece off, the next time we want to break off two pieces, and then four the next time, and then eight. *a* is one and *r* is 2.

#### Exercise 1

Determine whether the sequence is arithmetic, geometric, both, or neither.

If the sequence is arithmetic, find *d*. If the sequence is geometric, find *r*.

5, 5, 5, 5,...

#### Exercise 2

Determine whether the sequence is arithmetic, geometric, both, or neither.

If the sequence is arithmetic, find *d*. If the sequence is geometric, find *r*.

1, 2, 5, 10, 17,...

#### Exercise 3

Determine whether the sequence is arithmetic, geometric, both, or neither.

If the sequence is arithmetic, find *d*. If the sequence is geometric, find *r*.

81, -27, 9, -3,...

#### Exercise 4

Determine whether the sequence is arithmetic, geometric, both, or neither.

If the sequence is arithmetic, find *d*. If the sequence is geometric, find *r*.

100, 90, 80, 70, ...

#### Exercise 5

Determine whether the sequence is arithmetic, geometric, both, or neither.

If the sequence is arithmetic, find *d*. If the sequence is geometric, find *r*.

100, 50, -25, 12.5, 6.25, -3.125, ...

#### Exercise 6

Is it possible for a sequence to be both arithmetic and geometric?