Types of Sequences | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Types of Sequences

What types of sequences are there?

• Linear and quadratic sequences have their own specific notes.
• Other types of sequences include geometric and Fibonacci sequences, which are examined in more detail below.
• Additional sequences include cube numbers (cubic sequences) and triangular numbers.
• A common type of sequence in exam questions involves fractions combined with the above types.
  • Always look for features that make the position-to-term and term-to-term rules easy to identify.

What is a geometric sequence? 

• A geometric sequence can also be referred to as a geometric progression and sometimes as an exponential sequence
• In a geometric sequence, the term-to-term rule would be to multiply by a constant, r
  • a_n+1 = r.a_n
• r is called the common ratio and can be found by dividing any two consecutive terms, or
  • r = a_n+1 / a_n
• In the sequence 4, 8,  16,  32,  64, ... the common ratio, r,  would be 2 (8 ÷ 4 or 16 ÷ 8 or 32 ÷ 16 and so on)

What is a Fibonacci sequence? 

• The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
• The sequence begins with the first two terms as 1.
• Each subsequent term is the sum of the previous two terms.
  • The term-to-term rule is a_n+2 = a_n+1 + a_n
  • Two initial terms are needed to start a Fibonacci sequence.
• Any sequence with the term-to-term rule of adding the previous two terms is considered a Fibonacci sequence, even if the first two terms are not both 1.
• Fibonacci sequences often appear in nature, such as in the number of petals on flowers.

What is a cubic sequence?

• In a cubic sequence, the first differences between terms are not constant.
• The second differences (differences between the first differences) are also not constant.
• The third differences (differences between the second differences) are constant.
• Another way to understand this is that in a cubic sequence, the sequence of second differences forms a linear sequence.
• Example:
  • Sequence: 1, 5, 21, 55, 113, …
  • 1st Differences: 4, 16, 34, 58 (a Quadratic Sequence)
  • 2nd Differences: 12, 18, 24 (a Linear Sequence)
  • 3rd Differences: 6, 6, 6 (Constant)
• Constant third differences indicate a cubic sequence.

What should I be able to do with cubic sequence?

• You should be able to recognise and continue a cubic sequence
• You should also be able to find a formula for the nth term of a simple cubic sequence in terms of n
  • This formula will most likely be in one of the forms nth term = an³ or n³ + b
• To find the values of a and b, you must remember the terms in the sequence for n3 and compare them to the given sequence
  • n³ is the sequence 1, 8, 27, 64, 125, ....
  • Usually, each term will be either a little bit more or less than the sequence for n³
    • For example, the sequence 2, 9, 28, 65, 126, , ... has the formula n³ + 1 as each term is 1 more than the corresponding term in n³
  • Sometimes, each term will be two or three times the term in the sequence for n3
    • For example, the sequence 2, 16, 54, 128, 250, ... has the formula 2n³  as each term is twice the corresponding term in n³

Problem solving and sequences

• When the nature of the sequence is identified, it becomes feasible to determine unknown terms within the sequence. 
• This process often involves setting up and solving equations, which might include simultaneous equations. 
• Furthermore, some problems may entail sequences that correlate with well-known number sequences like square numbers, cube numbers, and triangular numbers.

Identifying Sequences

How do I identify a sequence?

• Is it clear from the given information? 
• Does the question explicitly state it? 
• Do you notice a consistent number that you use to transition from one term to the next? 
  • If there is, then it's likely a geometric sequence. 
• Then, observe the variances between the terms. 
  • If the first differences remain constant, it's indicative of a linear sequence. 
  • If the second differences remain constant, it's indicative of a quadratic sequence.

• Special cases to be aware of:
• If the differences repeat the original sequence
  • It is a geometric sequence with common ratio 2
• Fibonacci sequences also have differences that repeat the original sequence
  • However questions usually indicate if a Fibonacci sequence is involved

How do I continue a given sequence? 

Let's break down the approach to finding the next terms or filling in gaps within a sequence.

• CASE 1: Start by checking if there's a consistent difference between terms. If there is, it's a linear sequence. Simply add or subtract this common difference to each term to find the next ones. This approach follows the term-to-term rule.
• CASE 2: If there's no consistent difference, check if there's a common second or third difference. If so, it's a quadratic or cubic sequence respectively. Determine the next term by first identifying the next difference and then adding it to the sequence.
• CASE 3: If there's neither a common second nor third difference, inspect if there's a common multiplier between terms, indicating a geometric sequence. Find the next term by multiplying the last term by this common multiplier.
• CASE 4: Some sequences may not fit into these categories. In such cases, look for patterns in the differences. For instance, the differences may form a geometric sequence, which can help determine the next difference and subsequently, the next value in the sequence.
• In scenarios where you're asked to fill in gaps within a sequence, utilize the provided information to discern the sequence type and then proceed to complete the missing terms accordingly.