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Algebra (Intermediate) - Nth Terms and Sequences
 
Sequences | Nth Term | Quadratic Sequences
 
Sequences

A number sequence is a set of numbers (terms) in which a pattern can be seen and a rule can be used to find every term in the sequence.

For example:
  5,10,20,40… double the last term each time…..80,160

3,5,7,9,…..Add two each time…..11,13

25,21,17,13,…minus four each time…..9,5
 
The above examples have simple patterns, with harder sequences we need to look for a pattern and then establish the rule in order to calculate any term in the sequence.
 
 

Nth Term

The rule for finding any term is called the nth term.

For example:
Given the sequence 6,10,14,18,……
   
a) Find the nth term b) the 20th term c)If the nth term is 42, what is the value of n?
   
We look at the differences between each term
 
  6 10 14 18
\ _/\_ /\_ /
4 4 4		 The difference is four
   
  The general formula for the nth term is:
 
 
nth term = a + (n-1)d
where a= the first term=6
n = the number of the term
D= the difference = 4
 
 
For this sequence nth term = 6 +(n-1)4
= 6 + 4n -4
= 2 + 4n
     
We can now use this formula to work out the value of any term in the sequence.
     
     
b) 20th term =
2 + 4x20
= 82		 because n=20
     
c) nth term = 42  
     
  42 = 2 + 4n
40 = 4n
n=10		  
 

So the 10th term is 42.

This formula will work for any linear sequence. In a linear sequence the difference is constant. 4 in the sequence above.
     
Quadratic Sequences    
 
In this type the first difference is not constant. The second difference gives a constant.
 
For example:
3,8,15,24,36……… is a sequence.
  3 8 15 24 35
\_/\_ /\_ /\_ /
5 7 9 11
\_ /\_ /\_ /
2 2 2		 1st difference

2nd difference


This is a quadratic sequence as the 2nd difference is a constant (in this case, 2.)

The general formula for a quadratic is:
nth term = a + (n-1)d1 + ½(n-1)(n-2)d2
Where a = 1st term

d1= 1st difference

d2= 2nd difference
= 3 =5 = 2
 
  nth term = 3 + (n-1)5 + ½ (n-1)(n-2)2
=3 + 5n - 5 +n2 -3n + 2
=n2 +2n
 
We can use this to find the 100th term:
 
  100th term = 1002 +200
= 10200
 
Provided that the second difference is a constant we can use this method for any quadratic sequence.