TMUA Sequences & Series: AP, GP and Sums

Sequences and series feel deceptively friendly. The formulae are short, the patterns are familiar from GCSE, and most students assume a couple of plug-ins will see them through. The TMUA does not work like that. It takes the same arithmetic and geometric machinery you already know and hides it inside word problems, recurrence relations, sigma expressions and convergence conditions, then rewards whoever recognises the structure quickly and handles it without a calculator. This guide covers what is actually examined, arithmetic progressions, geometric progressions and the sum to infinity, sigma notation, and the traps that quietly cost marks.

What the TMUA actually examines

Sequences and series sit in the AS-level pure content and appear across the TMUA syllabus topics, most often as a standalone progression problem but sometimes woven into logarithms, inequalities or proof. The raw content is narrow: arithmetic progressions, geometric progressions, the sums of each, sigma notation, simple recurrence relations, and the convergence of a geometric series. There is no calculus of series and no induction on the syllabus, so the surface area is small.

What makes it examinable is disguise. You will rarely be handed a clean "find the tenth term" instruction. Instead a salary grows by a fixed amount each year and you must recognise an arithmetic progression, or a population multiplies by a constant factor and you must see a geometric one, or a sum is written in sigma notation and you must read off the first term and the common difference before any formula applies. The skill being tested is structural recognition followed by clean execution, which is exactly the kind of thing that makes the TMUA hard under time pressure. So the goal is not to learn more formulae, it is to make the four you need instant and to train your eye to spot which one is hiding.

Arithmetic progressions

An arithmetic progression, or AP, is a sequence where you add a fixed amount, the common difference $d$, to get from one term to the next. With first term $a$, the terms run $a, a+d, a+2d, a+3d, \dots$, and the nth term is

$u_n = a + (n-1)d.$

The $(n-1)$ rather than $n$ is the single most common slip, because the first term already sits there before any difference is added. Always sanity-check with $n = 1$: it must give back $a$.

The sum of the first $n$ terms has two equivalent forms. The one to memorise is

$S_n = \frac{n}{2}\left(2a + (n-1)d\right),$

and when you happen to know the last term $l$ it is often quicker to use $S_n = \frac{n}{2}(a + l)$, which just averages the first and last term and multiplies by how many there are. Both give the same number, so pick whichever matches the information you are given.

A typical TMUA dressing: the third term of an AP is $7$ and the eighth term is $22$. Find the sum of the first twenty terms. Subtracting the term equations gives $5d = 15$, so $d = 3$ and then $a = 1$. Feed those into the sum formula with $n = 20$ and you get $S_{20} = \frac{20}{2}(2 + 19 \times 3) = 10 \times 59 = 590$. Notice the work is mostly setting up two equations and solving them; the formula itself is the easy part. That is the pattern across the whole topic.

Geometric progressions and the sum to infinity

A geometric progression, or GP, is a sequence where you multiply by a fixed amount, the common ratio $r$, each step. With first term $a$, the terms run $a, ar, ar^2, ar^3, \dots$, and the nth term is

$u_n = ar^{n-1}.$

The sum of the first $n$ terms is

$S_n = \frac{a(1 - r^n)}{1 - r},$

valid for any $r \neq 1$. If $r$ is greater than one it is tidier to write it as $\frac{a(r^n - 1)}{r - 1}$ to keep the numbers positive, but it is the same formula. The thing examiners love most about GPs is the sum to infinity. When the common ratio is small enough that the terms shrink towards zero, the infinite sum settles on a finite value:

$S_\infty = \frac{a}{1-r}, \quad \text{valid only when } |r| < 1.$

That convergence condition is the heart of nearly every hard GP question. The series converges precisely when $-1 < r < 1$, in other words when the common ratio lies strictly between minus one and one. If $|r| \geq 1$ the terms do not shrink and the sum to infinity does not exist, so any question that asks for $S_\infty$ is implicitly telling you that $r$ is in that range, which is often the clue you need to solve for an unknown ratio.

A worked example. A GP has first term $12$ and sum to infinity $16$. Find the common ratio. Set $\frac{12}{1-r} = 16$, so $1 - r = \frac{12}{16} = \frac{3}{4}$, giving $r = \frac{1}{4}$. Check it lies in range: $\frac{1}{4}$ is between minus one and one, so the series genuinely converges and the answer is valid. Doing that range check is not optional politeness, it is how you reject the spurious solutions the examiners plant in the answer options. Keeping the whole calculation in exact fractions rather than decimals is part of the same calculator-free discipline the rest of the paper demands.

Try one

Reading about progressions only gets you so far. Here is a real past-paper question that runs on this machinery. Attempt it properly against the clock before revealing the solution, and watch how the progression is the skeleton of the problem rather than the whole of it:

If it felt awkward, it is almost always a recognition gap rather than a formula gap, and recognition is the cheapest thing to train.

Sigma notation and recurrence relations

Sigma notation is just shorthand for a sum, and the TMUA expects you to read it fluently. The expression $\sum_{k=1}^{n} u_k$ means add up the terms $u_k$ as the counter $k$ runs from $1$ to $n$. The skill is translation: given $\sum_{k=1}^{20} (3k + 1)$, recognise that plugging in $k = 1, 2, 3, \dots$ produces $4, 7, 10, \dots$, an arithmetic progression with first term $4$ and common difference $3$, then apply the AP sum formula. The sigma is a costume; underneath it is an AP or a GP you already know how to handle.

Two standard results help here. The sum of the first $n$ integers is $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$, and the sum of a constant is just that constant times how many terms there are, $\sum_{k=1}^{n} c = nc$. With those two you can split most linear sigma expressions into pieces you can evaluate by hand, which is faster than expanding the whole thing.

Recurrence relations define a sequence by telling you how to get the next term from the previous one, such as $u_{n+1} = u_n + 4$ with $u_1 = 3$. The move that unlocks these is recognising the hidden progression: a recurrence of the form $u_{n+1} = u_n + d$ is simply an AP with common difference $d$, and $u_{n+1} = r u_n$ is a GP with common ratio $r$. Once you have named it, every formula above applies. The TMUA likes recurrences precisely because they force you to do that recognition step rather than handing you the structure on a plate.

Common traps to avoid

A small set of errors accounts for most dropped marks, and the answer options are written to catch them.

• The off-by-one in the nth term. Writing $a + nd$ instead of $a + (n-1)d$, or $ar^n$ instead of $ar^{n-1}$. Always test with the first term: $n = 1$ must return $a$.
• Forgetting the convergence condition. Using the sum to infinity when $|r| \geq 1$, where it simply does not exist. If a question asks for $S_\infty$, the ratio must lie strictly between minus one and one, and you should verify any ratio you solve for.
• Confusing AP and GP. Adding when you should multiply, or vice versa. Read whether terms grow by a fixed amount or a fixed factor before reaching for a formula.
• Misreading sigma limits. A sum from $k = 0$ has one more term than a sum from $k = 1$, and a sum from $k = 3$ to $k = 10$ has eight terms, not seven. Count the terms deliberately.
• Decimal drift. Turning exact ratios like $\frac{1}{3}$ into rounded decimals, which corrupts the arithmetic in a no-calculator setting. Stay in fractions until the very end.

Most of these are recognition or care errors rather than deep maths, which is good news: they are the easiest kind to drill out. Lock the four core formulae and the convergence condition into memory, get fluent at spotting a disguised progression, and a topic that looks trivial but bites becomes a reliable source of quick marks. A structured plan that builds this fluency in early is laid out in how to prepare for the TMUA.

Frequently asked questions