Calculus How To

General Term: Definition, Examples

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Sequence and Series >

The general term (sometimes called the nth term) is a formula that defines a sequence.


For example, for the sequence defined by an = 1/n, the first four terms are found by plugging in 1, 2, 3, 4 for “n”:
1/1, 1/2, 1/3 and 1/4.
Different sequences have different formulas.

The general term is one way to define a sequence. The other way is the recursive definition of a sequence, which defines terms by way of other terms. For example, An = An-1 + 4.

General Term for Arithmetic Sequences

The general term for an arithmetic sequence is an = a1 + (n – 1) d, where d is the common difference.

Example question: What is the general term of the sequence 2, 5, 8,…?
Solution:

  1. Find d by subtracting the second from first term: d = 5 – 2 = 3.
  2. Plug d into the general formula: an = a1 + (n – 1) 3
  3. Plug in the first term for a1: an = 2 + (n – 1) 3

The general term is an = 2 + (n – 1) 3

General Term for Geometric Sequences

For a geometric sequence, the formula is an = a1 rn – 1, where r is the common ratio.

Example question: What is the general term of the geometric sequence 8, 4, 2,…?
Solution:

  1. Find r the ratio of any two consecutive terms. I’ll use the second and third terms in this example: r = 2/4 = ½.
  2. Plug r into the general formula: an = a1 ½n – 1
  3. Plug in the first term for a1: an = 8 (½)n – 1

The general term is an = 8 (½)n – 1.

Using Algebra

If you don’t know what kind of sequence you have, you may have to use a little logic and your knowledge of algebra to get your terms into a workable form. For example, let’s say you’re asked to find the general term for the sequence
general term of a sequence

This becomes much easier to work with if you change the denominators into exponents (giving an exponential sequence):
general term for an exponential sequence

Which gives an = ½n

References

Steig, J. 1999. Sequences & Series – General terms. Retrieved January 20, 2021 from: http://www.mesacc.edu/~marfv02121/readings/seq_def/

CITE THIS AS:
Stephanie Glen. "General Term: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/general-term/
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