Generating Sequences
This topic covers the two primary ways to define a sequence of numbers: by relating each term to the one before it, or by linking each term directly to its position in the list. ESAT questions test your ability to generate terms, work backwards, and solve problems using these rules.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A sequence can be defined by a 'term-to-term' rule, which tells you how to get the next term from the current one. This is also known as a recurrence relation.
- To use a term-to-term rule, you must know a starting term, usually the first term, u1.e.g., un+1 = 2un + 1
- A sequence can also be defined by a 'position-to-term' rule, which gives the value of the term based on its position, n. This is also called the nth term formula.
- A position-to-term rule allows you to find any term in the sequence directly, without needing to calculate all the preceding terms.e.g., un = 3n - 5
- You may need to solve an equation to find the position 'n' of a given term value. Remember that 'n' must be a positive integer.
Formulae
un+1 = f(un) This notation represents a term-to-term rule. You need a starting term (e.g., u1) to generate the sequence.
un = f(n) This notation represents a position-to-term rule. Substitute the desired position 'n' (e.g., n=5 for the 5th term) to find its value.
Definitions
- Sequence
- An ordered list of numbers, called terms, that follow a specific rule.
- Term-to-term rule
- A rule that defines how to calculate the next term in a sequence using the value of the previous term(s). Example: un+1 = un + 4.
- Position-to-term rule
- A formula that calculates the value of any term in a sequence directly from its position number (n). Example: un = n2 + 3.
Worked example
A sequence is generated by the term-to-term rule un+1 = 2un - 7. The third term, u3, is 15. What is the value of the first term, u1?
- 1
The problem requires working backwards from u3 to u1.
Start by finding u2.
- 2
Rearrange the rule to find the previous term:
un = (un+1 + 7) / 2 - 3
Calculate u2 using u3 = 15:
u2 = (15 + 7) / 2 = 22 / 2 = 11 - 4
Now calculate u1 using u2 = 11:
u1 = (11 + 7) / 2 = 18 / 2 = 9
Answer: 9
Common mistakes
- ×Making simple arithmetic errors under pressure, especially with negative numbers or multi-step rules like 'double then subtract 5'. Always check your calculations.
- ×Confusing the position number (n) with the term's value (un). For un = 2n + 5, the 4th term (n=4) is 2(4)+5=13, not the term when the value is 4.
- ×Incorrectly finding a specific term. For a term-to-term rule like un+1 = un + 3, you cannot find the 10th term by simply adding 3 to 10; you must generate the sequence term by term or find the nth term formula first.
No-calculator tips
- ✓When a rule is like un+1 = Aun + B, and you need to work backwards, mentally rearrange to un = (un+1 - B) / A to find previous terms quickly.
- ✓For linear sequences (e.g., 'add 6 each time'), to check if a large number 'X' is a term, subtract the first term and see if the result is divisible by the common difference.
- ✓When substituting 'n' into a position-to-term formula like n2 - 3n, factorise to n(n-3) if possible. It can make multiplication easier, e.g., for n=12, it's 12 × (12-3) = 12 × 9 = 108.