An **exponential sequence** *e*(*n*) is a list of numbers that follows the formula

** e(n) = A^{n}**.

*A* is a real or complex number and n is the term (i.e. 1, 2, 3, …). If A is > 1, the sequence shows exponential growth and <1 will give exponential decay.

## Exponential Sequence Example

If A is a real number, then e(n) is called a *real sequence*. For example, if A is 3, then the first four terms in the sequence are:

- 3
^{1}= 3 - 3
^{2}= 9 - 3
^{3}= 27 - 3
^{4}= 81.

## Relationship to Geometric Sequences

All exponential sequences are geometric sequences, with a common ratio equal to the base of the exponent (Pike, 2021).

A geometric sequence is a list of terms, where the next term is obtained by multiplying by the same amount (a common ratio) to get the next term. The above sequence 3^{n} has 3, 9, 27, and 81 as the first four terms, each of which can be obtained by multiplying the term before it by 3:

- 3 * 3 = 9
- 9 * 3 = 27
- 27 * 3 = 81.

## Finding the nth Term of an Exponential Sequence

Finding a formula for an exponential sequence is quite involved and there isn’t a formula you can follow to find it. However, knowing that it behaves as a geometric sequence makes it a lot easier to find the nth term of the sequence. For example, consider the following question:

**Example question:** What is the next term in this exponential sequence?

The easiest way to answer this type of question isn’t to figure out what A^{n} is. Instead, you know it behaves like a geometric sequence, so look for the common ratio. A quick glance tells us that the denominator is multiplied by 4 each time (4 * 4 = 16 and 16 * 4 = 64). So the denominator for the next term in the sequence is 64 * 4 = 256, so the term is 1/256.

## References

Pike, S. (2021). Geometric Sequences. Retrieved January 20, 2021 from: http://www.mesacc.edu/~scotz47781/mat150/notes/sequences/Geometric_Sequences.pdf

**CITE THIS AS:**

**Stephanie Glen**. "Exponential Sequence: Definition, Formula & Examples" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/exponential-sequence/

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