about Sequence

Sequence

A sequence is a set of numbers arranged one after another.

a1, a2, a3 ,..., an

3, 6, 9,..., 3n

The numbers a1, a2 , a3 , ..., are called terms or elements of the sequence.

The subscript is the set of positive integers 1, 2, 3, ... The subscript indicates the place that a term occupies in the sequence.

The nth term is denoted by an.

Calculation of a Sequence

By the nth term

an is a criterion that allows us to calculate any term of the sequence.

Example

an= 2n − 1

a1 = 2 ·1 − 1 = 1

a2 = 2 ·2 − 1 = 3

a3= 2 ·3 − 1 = 5

a4 = 2 ·4 − 1 = 7

1, 3, 5, 7, ..., 2n −1

Not all sequences have a general term. For example, the sequence of prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23,...

By a Recursive Formula

A term is obtained by operating with the previous terms.

Example

Write a sequence whose first term is 2, knowing that each term is the square of the previous term.

2, 4, 16, ...

Fibonacci Sequence or Fibonacci Number

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...

The first two terms are one and the other terms are obtained by adding the two previous terms.

nth Term

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

2.Check 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

3.Check 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

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

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

an= n3

5.Check 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

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

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