June 26, 2019

## 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.

a_{n}= 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.

a_{n} = 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.

b_{n}= 2 + (n − 1) · 1 = 2 + n −1 = n+1

a_{n}= (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.

a_{n}= (n + 1) ² + 1

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

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

a_{n}= (n + 1)² + 2

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

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

a_{n}= (n + 1)² − 1

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

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

a_{n}= (n + 1) ² − 2

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

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

a_{n}= n³

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

If the odd terms are negative and the even terms are positive, multiply **a _{n}** by

**(−1)**.

^{n}−4, 9, −16, 25, −36, 49, ...

a_{n}= (−1)^{n} (n + 1)²

If the odd terms are positive and the even terms are negative, multiply **a _{n}** by

**(−1)**or

^{n+1 }**(−1)**.

^{n−1}4, −9, 16, −25, 36, −49, ...

a_{n}= (−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.

a_{n}= b_{n} /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.

a_{n}= (3n − 1)/(n + 1)²