GCSE Maths · Sequences and nth Term · Higher · Summary

GCSE Maths Sequences and nth Term (Higher): summary and key words

Chapter summary

Skill	Method
Term-to-term rule	Find the differences (add/subtract), then the ratios (multiply/divide); apply the rule to the last term
Position-to-term rule	Replace n by the position and follow BIDMAS; the first term is n = 1
Pattern sequences	Count each pattern, find the rule, explain it with the picture, continue the sequence
nth term of an arithmetic sequence	Find d, write the d times table, subtract to find the adjustment: dn + (first term − d)
Is a number in the sequence?	Solve nth term = number; it is a term only if n is a positive whole number
First term above (below) a value	Solve an inequality; round n up (down) to a whole number and test the term before
Special sequences	Squares n², cubes n³, triangular n(n + 1) ÷ 2, powers of 2: 2ⁿ, Fibonacci-type: add the previous two
Geometric sequences	Common ratio r = term ÷ previous term; powers rⁿ; Higher: nth term ar^{n − 1}; surds: √k × √k = k
Two terms given (Higher)	d = (change in terms) ÷ (change in positions), then find the adjustment
Quadratic sequences (Higher)	Second differences constant; a = half the second difference; subtract an²; find the linear part bn + c
Proving with sequences	Use the nth term and algebra, e.g. 6n + 3 = 3(2n + 1)

I can continue a sequence and describe its term-to-term rule in words.
I can find terms from a position-to-term rule, including powers such as n² and 2ⁿ.
I can continue patterns made from matchsticks, tiles and dots.
I can find the nth term of an arithmetic sequence.
I can decide whether a number is in a sequence and find the first term above or below a value.
I can recognise square, cube, triangular, Fibonacci-type and geometric sequences.
I can find the common ratio of a geometric sequence and continue it.
Higher: I can find the nth term of a quadratic sequence using second differences.
Higher: I can find the nth term of a geometric sequence and work with a surd ratio.
I can use the nth term to explain or prove something about every term.

Sequence: A list of numbers that follow a rule.
Term: One number in a sequence.
Term-to-term rule: A rule that tells you how to get from one term to the next.
Position-to-term rule (nth term): A formula that gives the term from its position n.
Position: The place of a term in the sequence: 1st, 2nd, 3rd, …
Arithmetic (linear) sequence: A sequence that goes up or down by the same amount each time.
Common difference: The amount added each time in an arithmetic sequence.
Geometric sequence: A sequence made by multiplying by the same number each time.
Common ratio: The number multiplied each time in a geometric sequence.
Square number: A number multiplied by itself: 1, 4, 9, 16, …
Cube number: A number multiplied by itself twice more: 1, 8, 27, 64, …
Triangular number: The numbers 1, 3, 6, 10, 15, … found by adding 1, then 2, then 3, …
Fibonacci-type sequence: A sequence where each term is the sum of the two terms before it.
First differences: The differences between neighbouring terms.
Second differences: The differences between the first differences.
Quadratic sequence: A sequence whose nth term contains n²; its second differences are constant (Higher).
Surd: A root that is not exact, such as √2 (Higher).