Common Number Sequences

Natural numbers

The sequence of positive integers. The difference between the nth term and the previous term is 1 and the first term is 1

 
nth term
First term:

1
nth term:

n
(n-1)th term:

n - 1
Term difference:

1
Sequence type:

Arithmetic: a = 1, d = 1
Iterative definition:

u1 = 1, un = un-1 + 1 for all n ≥ 2
Sequence:

1 2 3 4 5 6 ... n
Sum of the first n terms:

n(n + 1)
2

Generate values

Enter a value of n to find for the sequence of natural numbers.

  • The nth term
  • The sum of the first n terms

n

Odd Numbers

The sequence of positive integer that are not divisible by two.

 
nth term
First term:

1
nth term:

2n - 1
(n-1)th term:

2n - 3
Term difference:

2
Sequence type:

Arithmetic: a = 1, d = 2
Iterative definition:

u1 = 1, un = un-1 + 2 for all n ≥ 2
Sequence:

1 3 5 7 9 1... (2n - 1)
Sum of the first n terms:

n2

Generate values

Enter a value of n to find for the sequence of odd numbers.

  • The nth term
  • The sum of the first n terms

n

Even Numbers:

The sequence of positive integer that are divisible by two.

 
nth term
First term:

2
nth term:

2n
(n-1)th term:

2n - 2
Term difference:

2
Sequence type:

Arithmetic: a = 2, d = 2
Iterative definition:

u1 = 2, un = un-1 + 2 for all n ≥ 2
Sequence:

2 4 6 8 10 12 ... 2n

2
Sum of the first n terms:

n(n + 1)

Generate values

Enter a value of n to find for the sequence of odd numbers.

  • The nth term
  • The sum of the first n terms

n

Square Numbers

A sequence where the nth term is equal to n × n

 
nth term
First term:

1
nth term:

n2
(n-1)th term:

(n - 1)2
Term difference:

2n - 1 (nth odd number)
Sequence type:

Power: r = 2
Iterative definition:

u1 = 1, un = un-1 + 2n - 1 for all n ≥ 2
Sequence:

1 4 9 16 25 36 ... n2
Sum of the first n terms:

n(n + 1)(2n + 1)
6

Generate values

Enter a value of n to find for the sequence of square numbers.

  • The nth term
  • The difference between the nth and the previous term
  • The sum of the first n terms
  • A list of the first n terms

n

Cubic numbers

A sequence where the nth term is equal to n × n × n or n3

 
nth term
First term:

1
nth term:

n3
(n-1)th term:

(n - 1)3
Term difference:

3n2 - 3n + 1
Sequence type:

Power: r = 3
Iterative definition:

u1 = 1, un = un-1 + 2n - 1 for all n ≥ 2
Sequence:

1 8 27 64 125 216 ... n3
Sum of the first n terms:

n2(n+1)2
4

Generate values

Enter a value of n to find for the sequence of cubic numbers.

  • The nth term
  • The difference between the nth and the previous term
  • The sum of the first n terms
  • A list of the first n terms

n

Triangular numbers:

A sequence where the difference between the nth term and the previous term is n and the first term is 1

 
nth term
First term:

1
nth term:

n(n+1)
2
(n-1)th term:

n(n-1)
2
Term difference:

n
Iterative definition:

u1 = 1, un = un-1 + n for all n ≥ 2
Sequence:

1 3 6 10 15 21 ...
n(n+1)
2
Sum of the first n terms:

n(n+1)(n+2)
6

Generate values

Enter a value of n to find for the sequence of triangular numbers.

  • The nth term
  • The difference between the nth and the previous term
  • The sum of the first n terms
  • A list of the first n terms

n

Factorial numbers

A sequence where the nth term is the product of all of the integers between 1 and n

 
nth term
First term:

1
nth term:

n! = (n × (n-1) × (n-2) × ... × 1)
(n-1)th term:

(n-1)!
Iterative definition:

u1 = 1, un = un-1 × n for all n ≥ 2
Sequence:

1 2 6 24 120 720 ... n!

Generate values

Enter a value of n to find for the sequence of factorial numbers.

  • The nth term
  • A list of the first n terms

n

Powers of 2

A sequence where the nth term is equal to 2(n-1).

 
nth term
First term:

1
nth term:

2n-1
(n-1)th term:

2n-2
Sequence type:

Geometric: a = 1, r = 2
Iterative definition:

u1 = 1, un = un-1 × 2 for all n ≥ 2
Sequence:

1 2 4 8 16 32 ... 2n-1
Sum of the first n terms:

2n - 1

Generate values

Enter a value of n to find for the sequence of powers of 2.

  • The nth term
  • The difference between the nth and the previous term
  • The sum of the first n terms
  • A list of the first n terms

n

Prime numbers:

A sequence (in ascending order) of integers greater than one that cannot be expressed as the product of two smaller integer values.

 
nth term
First term:

2
nth term:

There is no formula for the nth prime
(n-1)th term:

Generate values

Enter a value of n to find for the sequence of prime numbers.

  • The nth term
  • A list of the first n terms

n

Fibonacci numbers

A sequence where the nth term is the sum of the two previous terms, and the first two terms are 1 and 1.

 
nth term
First two terms:

1, 1
nth term

1
√5

((

1 + √5
2

)n

- (

1 - √5
2

)n)

1
(n-1)th term:

 
(n-2)nd term:

 
Term difference:

(n-2)th term
Iterative definition:

u1 = 1, u2 = 1, un = un-1 + un-2 for all n ≥ 3
Sequence:

u1 = 1, u2 = 1, un = un-1 + un-2 for all n ≥ 3 ... 2n-1
Sum of first n terms:

1
√5

((

1 + √5
2

)n+2

- (

1 - √5
2

)n+2) - 1 or un+2 - 1

Generate values

Enter a value of n to find for the sequence of Fibonacci numbers.

  • The nth term
  • The difference between the nth and the previous term
  • The sum of the first n terms
  • A list of the first n terms

n