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Common Sequences and Series
Common Sequences and Series:
Types of Sequences
Arithmetic Sequence:
A sequence where each pair of terms has a common difference.
Example: 4 7 10 13 16 ...
nth term
First term:
a
Common difference:
d
Sequence:
a a+d a+2d a+3d ...
nth term:
a + (n - 1)d
Iterative definition:
u
1
= a, u
n
= u
n-1
+ d for all n ≥ 2
Sum of first n terms:
n(2a + (n - 1)d)
2
Find values for given values of a, d and n
a
d
n
Geometric Sequence:
A sequence where each pair of terms has a common ratio
Example: 3 6 12 24 48 ...
nth term
First term:
a
Common difference:
r
Sequence:
a ar ar
2
ar
3
...
nth term:
ar
n-1
Iterative definition:
u
1
= a, u
n
= u
n-1
× r for all n ≥ 2
Sum of first n terms:
a(r
n
-1)
r - 1
Series behaviour:
The series diverges for |r| ≥ 1 and converges for |r| < 1
Sum of infinite terms:
a
1 - r
Find values for given values of a, r and n
a
r
n
Power Sequence:
The sequence of positive integers where the nth term is equal to n
r
for a given positive integer r
nth term
First term:
1
nth term:
n
r
Sequence:
1
r
2
r
3
r
4
r
5
r
6
r
...
Find values for a given value of n
Find values for given values of r and n
r
n
Periodic Sequence:
A sequence which repeats after every p terms where p is the period
Example: 4 7 3 1 4 7 3 1 ... (period 4)
nth term
Period:
p
1
Initial terms:
u
1
Sequence posision:
k where k = ((n-1) mod p) + 1
nth term:
u
n
= u
k
Iterative definition:
u
1
u
2
, ... u
p
given. u
n
= u
n - p
for n > p
Sum of first n terms:
(u
1
+ ... + u
p
) × q + (u
1
+ ... + u
k
) where q = n/p rounded down
Find values for given values of p and n given the first p terms are the first p natural numbers
p
n
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:
u
1
= 1, u
n
= u
n-1
+ 1 for all n ≥ 2
Sequence:
1 2 3 4 5 6 ...
Sum of first n terms:
n(n + 1)
2
Find values for a given value of n
n
Odd numbers:
The sequence of positive integer that are not divisible by 2.
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:
u
1
= 1, u
n
= u
n-1
+ 2 for all n ≥ 2
Sequence:
1 3 5 7 9 11 ...
Sum of first n terms:
n
2
Find values for a given value of n
n
Even numbers:
The sequence of positive integer that are divisible by 2.
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:
u
1
= 2, u
n
= u
n-1
+ 2 for all n ≥ 2
Sequence:
2 4 6 8 10 12 ...
Sum of first n terms:
n(n + 1)
Find values for a given value of n
n
Square numbers:
A sequence where the nth term is equal to n × n
nth term
First term:
1
nth term:
n
2
(n-1)th term:
(n - 1)
2
Term difference:
2n - 1 (nth odd number)
Sequence type:
Power: r = 2
Iterative definition:
u
1
= 1, u
n
= u
n-1
+ 2n - 1 for all n ≥ 2
Sequence:
1 4 9 16 25 36 ...
Sum of first n terms:
n(n + 1)(2n + 1)
6
Find values for a given value of n
n
Cubic numbers:
A sequence where the nth term is equal to n × n × n or n
3
nth term
First term:
1
nth term:
n
3
(n-1)th term:
(n - 1)
3
Term difference:
3n
2
- 3n + 1
Sequence type:
Power: r = 3
Iterative definition:
u
1
= 1, u
n
= u
n-1
+ 2n - 1 for all n ≥ 2
Sequence:
1 8 27 64 125 216 ...
Sum of first n terms:
n
2
(n+1)
2
4
Find values for a given value of n
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:
u
1
= 1, u
n
= u
n-1
+ n for all n ≥ 2
Sequence:
1 3 6 10 15 21 ...
Sum of first n terms:
n(n+1)(n+2)
6
Series behaviour:
The series diverges
Find values for a given value of n
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:
u
1
= 1, u
n
= u
n-1
× n for all n ≥ 2
Sequence:
1 2 6 24 120 720 ...
Find values for a given value of n
n
Powers of 2:
A sequence where the nth term is equal to 2
(n-1)
nth term
First term:
1
nth term:
2
n-1
(n-1)th term:
2
n-2
Sequence type:
Geometric: a = 1, r = 2
Iterative definition:
u
1
= 1, u
n
= u
n-1
× 2 for all n ≥ 2
Sequence:
1 2 4 8 16 32 64 128 ...
Sum of first n terms:
2
n
- 1
Find values for a given value of n
n
Prime numbers:
A sequence (in ascending order) of integers greater than 1 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:
Sequence:
2 3 5 7 11 13 17 19 ...
Number of terms:
There are infinitely many prime numbers
Find values for a given value of n
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:
n(n+1)
2
nth term
1
√5
((
1 + √5
2
)
n
- (
1 - √5
2
)
n
)
(n-1)th term:
(n-2)nd term:
Term difference:
(n-2)th term
Iterative definition:
u
1
= 1, u
2
= 1, u
n
= u
n-1
+ u
n-2
for all n ≥ 3
Sequence:
1 1 2 3 5 8 13 22 ...
Sum of first n terms:
1
√5
((
1 + √5
2
)
n+2
- (
1 - √5
2
)
n+2
) - 1 or u
n+2
- 1
Series behaviour:
The series diverges
Find values for a given value of n
n
