SequencesEdexcel IGCSE Maths: Flashcards

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What is a term-to-term rule?

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What is a term-to-term rule?: A rule that describes how to get the next term in a sequence from the previous term(s), e.g. "add 5 to the previous term".
What is a position-to-term rule (nth term)?: An algebraic formula in terms of $n$ that gives the value of any term directly from its position, without needing earlier terms.
What is the nth term formula for a linear sequence with first term $a$ and common difference $d$ ?: $u_n = dn + (a-d)$
Sequence: $4, 9, 14, 19, \dots$ Find the nth term.: Common difference $d=5$ ; nth term $=5n-1$
How do you recognise a quadratic sequence?: The first differences are not constant, but the second differences (differences of the differences) are constant.
How do you find the coefficient $a$ of $n^2$ in a quadratic nth term?: Divide the constant second difference by 2: $a = \dfrac{\text{second difference}}{2}$
Sequence: $3, 6, 11, 18, 27$ . What are the first and second differences?: First differences: $3, 5, 7, 9$ . Second differences: $2, 2, 2$ (constant, so quadratic).
What is the nth term of the square numbers?: $n^2$ , giving $1, 4, 9, 16, 25, \dots$
What is the nth term of the triangular numbers?: $\dfrac{n(n+1)}{2}$ , giving $1, 3, 6, 10, 15, \dots$
What defines a Fibonacci-type sequence?: Each term is the sum of the two preceding terms, e.g. $1, 1, 2, 3, 5, 8, \dots$
What is the nth term formula for a geometric progression?: $u_n = ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
How do you find the common ratio $r$ of a geometric progression?: Divide any term by the term immediately before it: $r = \dfrac{u_{n+1}}{u_n}$ .
Sequence: $2, 6, 18, 54, \dots$ Find $r$ and the nth term.: $r=3$ ; nth term $= 2(3)^{n-1}$
Can the common ratio of a geometric progression be negative?: Yes — if $r$ is negative, the terms alternate in sign.
What is the key difference between how you analyse arithmetic vs geometric sequences?: For arithmetic/linear sequences you subtract consecutive terms to find $d$ ; for geometric progressions you divide consecutive terms to find $r$ .