Sequences and Series

Overview

Sequences and series turn a pattern into mathematics you can test, generalise and use. In WJEC 2.3.4, candidates move between individual terms, totals, sigma notation, recurrence relations and binomial expansions with rational powers. The topic rewards disciplined method: identify the pattern before choosing a formula, substitute with brackets, and check any condition before claiming a result.

This content is highly examinable because one idea often changes form within a single question. A geometric model may lead to a sum to infinity; a recurrence can describe an iterative process; a binomial expansion can produce a numerical approximation. Questions commonly ask candidates to show, find, use, prove, state the range of validity or model. Marks are awarded for a correct formula, accurate substitution, algebraic control and an appropriate conclusion. It connects directly to exponential growth and decay, numerical methods, functions, proof and approximation.

Key concepts

1. Sequence, series and recurrence: know what the question is asking

A sequence is an ordered list of terms. The notation (u_n) means “the term in position (n)”. A series is what you obtain when you add sequence terms; its partial sum is written (S_n). Do not use a sum formula merely because the word “series” appears: first decide whether the question asks for one term or a total.

A sequence can be supplied explicitly, for example (u_n=3n-1), or recursively, for example (x_{n+1}=0.6x_n+4) with a starting value. In a recurrence, calculate terms in order. The subscript is not a power: (x_{n+1}) is the next term, not (x) to a power.

Classification test: subtract consecutive terms. If the difference is constant, the sequence is arithmetic. If it is not, divide consecutive non-zero terms. If the ratio is constant, it is geometric. For example, (5,9,13,17) has common difference (d=4); (5,10,20,40) has common ratio (r=2). State this identification when it justifies your formula. It makes the method visible to the examiner and prevents a formula swap.

2. Arithmetic sequences and series: add the same amount

An arithmetic sequence changes by a fixed common difference (d). Its first term is (a), so the nth term is

[u_n=a+(n-1)d.]

The (n-1) matters. From term 1 to term (n), there are (n-1) jumps of size (d). For the sequence (7,12,17,\ldots), (a=7) and (d=5), so (u_{20}=7+19(5)=102), not (107). A common lost mark comes from writing (a+nd), which starts one jump too far.

For a total of the first (n) terms, use either

[S_n=\frac{n}{2}[2a+(n-1)d]\qquad\text{or}\qquad S_n=\frac{n}{2}(a+l),]

where (l) is the last term. The second form is efficient if the question has already found (u_n). Both expressions work because each first-and-last pair has the same total: (a+l). There are (n/2) such pairs, even when (n) is odd, because the pairing is an algebraic argument rather than a physical rearrangement.

Exam trigger: when the wording says “total”, “sum of the first”, “how many altogether” or “accumulated”, write (S_n), not (u_n). Keep (a), (d), (n) and (l) labelled on a line before substitution. This supports method marks if arithmetic later slips.

3. Geometric sequences and series: multiply by the same factor

A geometric sequence has a fixed common ratio (r). Its nth term is

[u_n=ar^{n-1}.]

Find (r) by division in the forward direction: for (81,27,9,3), (r=27/81=1/3), not (3). A negative ratio gives alternating signs, such as (6,-3,1.5,-0.75,\ldots), where (r=-1/2).

The finite sum is

[S_n=\frac{a(1-r^n)}{1-r}\quad (r
e1).]

Brackets are essential, especially if (r) is negative. The formula comes from subtracting (rS_n) from (S_n), causing the middle terms to cancel. This cancellation is why geometric sums have a compact formula while arbitrary series do not.

An infinite geometric series has a finite sum only when (|r|<1):

[S_\infty=\frac{a}{1-r}.]

The condition is not decoration. If repeated multiplication makes terms approach zero, the leftover tail becomes negligible; if (|r|\geq1), it does not. For (10+5+2.5+\cdots), (r=1/2), so (S_\infty=10/(1-1/2)=20). For (10+15+22.5+\cdots), (r=1.5), so a sum to infinity is invalid. Credit is given for explicitly checking (|r|<1) before using the formula.

4. Sigma notation, modelling and iteration

Sigma notation is compressed addition. (\sum_{r=1}^{4}(2r-1)) means substitute (r=1,2,3,4) and add the four resulting terms: (1+3+5+7). The lower limit tells you where to start; the upper limit tells you where to stop. Write out the first few terms where the limits or indexing could cause an error.

Use sigma notation to define a sum exactly, for example (\sum_{r=1}^{n}[a+(r-1)d]) for an arithmetic series. A changing letter such as (r) is a local index, not automatically the geometric common ratio. Read the notation from the limits outward.

Recurrences model “next value depends on current value” situations: a population after a percentage change, a balance with regular adjustment, or an iterative numerical approximation. Calculate sequentially and retain enough accuracy. Ask whether the values are increasing, decreasing, periodic or tending to a limiting value. A sensible model needs an interpretation: negative population, impossible probability or unbounded “remaining amount” may reveal that the model no longer applies.

5. Binomial expansion with rational indices: factor, expand, restrict

For a rational or negative index, first convert the expression to ((1+x)^n) form. The formula booklet gives

[(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots]

for rational (n), valid when (|x|<1). More generally, factor (a) from ((a+bx)^n):

[(a+bx)^n=a^n\left(1+\frac{b}{a}x\right)^n,]

so the validity condition is (\left|\frac{b}{a}x\right|<1). Do not silently treat ((4-x)^{1/2}) as ((1-x)^{1/2}). Factor it first: (2(1-x/4)^{1/2}).

For non-integer (n), the expansion generally continues forever. “Up to and including the term in (x^2)” means stop after the (x^2) term, not after two terms. Work coefficient by coefficient, retaining signs carefully. For (n=1/2), the coefficient of (x^2) is ((1/2)(-1/2)/2=-1/8), so it is negative. State the range of validity as a separate final line. A correct expansion without its condition is incomplete where the question asks for it.

Binomial series are also tools for approximation. Choose a nearby value so that the new inner (x) is small. For example, (\sqrt{3.96}=2\sqrt{0.99}=2(1-0.01)^{1/2}). Successive terms shrink quickly because (|-0.01|<1). Give an approximation sign, not an equality sign, once the series has been truncated.

Formula and relationship bank

Graph and data skills

A sequence is discrete: plot individual points ((n,u_n)), not a continuous curve, unless the question asks you to compare it with a function. Arithmetic terms lie on a straight-line pattern against (n); geometric terms show exponential growth, exponential decay or alternating behaviour. When a recurrence is modelled in a table, label the index and retain calculator accuracy until the requested final rounding. For a limiting value, inspect successive terms and justify the behaviour from the recurrence or ratio rather than merely stating that a calculator display “looks stable”.

Practical applications

Arithmetic models a fixed additional amount each period: stair risers, a saving plan with a constant extra deposit, or seating arranged with the same number added per row. Geometric models repeated proportional change: depreciation, compound growth, bouncing-ball distances and signal decay. Binomial series gives quick approximations when a calculator value is close to a simple known value. In each setting, define (a), (d) or (r) in context and check whether an infinite model is physically reasonable. A ball may bounce indefinitely in a model, but a real measurement system has a detection limit.

Listen and recall

Use the tutor-led podcast for a ten-minute first pass, then cover the formula bank and answer the retrieval cues from memory.