nth Term

Common Number Patterns

A sequence of numbers can have a unique pattern. To find the most common patterns and how they are made, calculate the nth term of the sequence.

Calculating the Nth Term of a Sequence

To determine the nth term, follow these steps:

1 Check if the sequence is an arithmetic sequence.

8, 3, −2, −7, −12, ...

3 − 8 = −5

−2 − 3 = −5

−7 − (−2) = −5

−12 − (−7) = −5

d = −5.

an= 8 + (n − 1) (−5) = 8 −5n +5 = −5n + 13


2Check if the sequence is a geometric sequence.

3, 6, 12, 24, 48, ...

6/3 = 2

12/6 = 2

24/12 = 2

48/24 = 2

r= 2.

an = 3· 2 n−1


3Check if the terms of the sequence are square numbers.

4, 9, 16, 25, 36, 49, ...

22, 32, 42, 52, 62, 72, ...

Note that the bases are in an arithmetic sequence, where d = 1, and the exponent is a constant.

bn= 2 + (n − 1) · 1 = 2 + n −1 = n+1

an= (n + 1)2

Also, sequences whose terms are numbers next to perfect squares can be found.

5, 10, 17, 26, 37, 50, ...

22 + 1 , 32 + 1, 42 + 1, 52 + 1, 62 + 1 , 72 + 1, ...

Find the nth term as in the previous example and add 1.

an= (n + 1) 2 + 1

6, 11, 18, 27, 38, 51, ...

22 + 2, 32 + 2, 42 + 1, 52 + 2, 62 +2, 72 + 2, ...

an= (n + 1)2 + 2

3, 8, 15, 24, 35, 48, ...

22 − 1, 32 − 1, 42 −1, 52 − 1, 62 − 1, 72 − 1, ...

an= (n + 1)2 − 1

2, 7, 14, 23, 34, 47, ...

22 −2 , 32 −2, 42 −2, 52 −2, 62 −2 , 72 −2, ...

an= (n + 1) 2 − 2


4Check if the terms of the sequence are cube numbers.

1, 8, 27, 64, 125, 216, 343, ...

an= n3


5Check if the terms of the sequence change sign consecutively.

If the odd terms are negative and the even terms are positive, multiply an by (−1)n.

−4, 9, −16, 25, −36, 49, ...

an= (−1)n (n + 1)2

If the odd terms are positive and the even terms are negative, multiply an by (−1)n+1 or (−1)n−1.

4, −9, 16, −25, 36, −49, ...

an= (−1)n+1 (n + 1)2


6Check if the terms of the sequence are fractional and it is not an arithmetic or geometric sequence.

Calculate the nth term of the numerator and denominator separately.

an= bn /c n

2/4, 5/9, 8/16, 11/25, 14/36,...

There are two sequences:

2, 5, 8, 11, 14, ...

4, 9, 16, 25, 36, ...

The first is an arithmetic sequence with d = 3 and the second is a sequence of perfect squares.

an= (3n − 1)/(n + 1)2