Exponential Sequence: Definition, Formula & Examples

Sequence and Series >

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

e(n) = Aⁿ.

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:

• 3¹ = 3
• 3² = 9
• 3³ = 27
• 3⁴ = 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 3ⁿ 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 Aⁿ 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

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