Number and Algebra: Sequences, Polynomials & the Marks Examiners Actually Drop

Q: What's the difference between arithmetic and geometric sequences?

Arithmetic sequences add a constant difference $d$ each step (linear growth); geometric sequences multiply by a constant ratio $r$ each step (exponential growth or decay).

Q: When does a geometric series have a sum to infinity?

Only when $|r|<1$, so the terms shrink towards zero. The formula is $S_\infty=\frac{u_1}{1-r}$, and you must state the condition to get full marks.

Q: Is proof by induction tested at SL?

No, induction is an HL-only technique in IB Maths AA. It's used to prove general formulas for sequences and other statements involving all positive integers.

Q: How do I know if a rational function has a vertical asymptote or a hole?

Fully factorise the numerator and denominator. If a factor is common to both, it cancels and creates a hole; any uncancelled denominator zero gives a vertical asymptote.

Q: What are Vieta's formulas used for?

They let you find combinations of a polynomial's roots (like their sum or product) directly from its coefficients, without solving the equation.

Q: Why is the remainder theorem useful?

It lets you find the remainder of $P(x)\div(x-a)$ by simply substituting $x=a$ into $P(x)$, avoiding long division when you don't need the quotient.

The five ideas behind almost every Paper 1 and Paper 2 algebra question in IB Maths AA.

Diagram linking arithmetic and geometric sequences, induction, polynomials and rational function graphs

Subject

Maths AA

Curriculum

IB Diploma Programme

Grade

Topic

Number and Algebra

Reading

7 min

Difficulty

Advanced

Quick facts

Difficulty

★★★★☆

Exam weight

~1/5 of AA HL Paper 1 + Paper 2 combined

Prerequisites

Basic algebra, function notation, simple equation solving

You'll learn

Sequences, series, induction, roots of polynomials, asymptotes

Revision time

45–60 min for full note

Number and Algebra ties together some of the most exam-frequent ideas in IB Maths AA: arithmetic and geometric sequences, the sum to infinity, proof by induction, and the algebra linking a polynomial's coefficients to its roots. Sequences and series appear almost every year in both Paper 1 and Paper 2, often stacked with 'show that' derivations or GDC-heavy equations. Polynomials and rational functions extend the same logic to continuous graphs — where roots become factors and asymptotes reveal a function's long-term behaviour. This teaser walks through the five concepts that generate the most marks (and the most dropped marks): sequence formulas, sum to infinity, induction proofs, Vieta's formulas, and asymptote rules. The full revision note covers every formula, worked trap, and mark scheme detail in depth.

What you’ll be able to do

✓Apply the nth term and sum formulas for arithmetic sequences

✓Apply the nth term, sum, and sum to infinity formulas for geometric sequences

✓State and use the convergence condition for a geometric series

✓Construct a complete proof by mathematical induction (HL)

✓Use the remainder and factor theorems to factorise polynomials

✓Use Vieta's formulas to find combinations of roots without solving the equation

✓Identify vertical, horizontal, and oblique asymptotes of a rational function

✓Avoid the most common mark-losing traps in 'show that' and induction questions

Arithmetic Sequences: Constant Difference

An arithmetic sequence adds (or subtracts) the same amount $d$ every step — think a car depreciating by a fixed dollar amount per year. The nth term is $u_{n} = u_{1} + (n - 1) d$ , and note it's $(n - 1)$ lots of $d$ , not $n$ , since $u_{1}$ already contains zero jumps. When a question gives you two terms or two sums, expect simultaneous equations to find $u_{1}$ and $d$ .

Number line showing an arithmetic sequence with equal jumps of size d

Exam tip

Once you know the last term, use $S_{n} = \frac{n}{2} (u_{1} + u_{n})$ instead of re-deriving $d$ — it's faster and avoids sign errors.

Mini summary

Constant difference, linear growth — $u_{n} = u_{1} + (n - 1) d$ .

Geometric Sequences and the Sum to Infinity

A geometric sequence multiplies by a constant ratio $r$ each step — a ball rebounding to a fraction of its drop height, or compound interest. The nth term is $u_{n} = u_{1} r^{n - 1}$ , and the sum of the first $n$ terms can be written two ways: $S_{n} = \frac{u _{1} ( r ^{n} - 1 )}{r - 1} = \frac{u _{1} ( 1 - r ^{n} )}{1 - r}$ . If $∣ r ∣ < 1$ , the terms shrink to nothing and $S_{\infty} = \frac{u _{1}}{1 - r}$ — the only formula here with a condition attached.

Bouncing ball diagram showing decreasing rebound heights forming a geometric sequence

Feature	Arithmetic	Geometric
Pattern	Constant difference $d$ (add/subtract)	Constant ratio $r$ (multiply)
Growth type	Linear	Exponential
nth term	$u_{n} = u_{1} + (n - 1) d$	$u_{n} = u_{1} r^{n - 1}$
Sum to infinity	Never exists	Exists only if $∣ r ∣ < 1$

Common mistake

Doubling the very first drop height in a bouncing-ball question — only bounces after the first impact go up-and-down, the initial drop is single-direction.

Mini summary

Constant ratio, exponential growth or decay — $S_{\infty}$ only exists for $∣ r ∣ < 1$ .

Proof by Mathematical Induction (HL)

Induction is how examiners force you to justify a general formula for a sequence rather than just quoting it. A complete proof needs four separately mark-worthy parts: a base case, an explicit assumption for $n = k$ , algebra showing the result holds for $n = k + 1$ , and a concluding sentence tying it all together.

Flowchart of the four steps of a proof by mathematical induction

Common mistake

Missing the base case or skipping the final concluding sentence — each costs a mark independently of whether the algebra is correct.

Mini summary

Base case, assumption, $n = k + 1$ step, conclusion — all four are graded separately.

Remainder Theorem, Factor Theorem & Vieta's Formulas

The remainder theorem says dividing $P (x)$ by $(x - a)$ leaves remainder $P (a)$ — substitute, don't divide, if that's all you need. The factor theorem is the special case where $P (a) = 0$ , confirming $(x - a)$ is a factor. Once roots are known, Vieta's formulas extract combinations of roots directly from the coefficients: for a cubic, $α + β + γ = - \frac{b}{a}$ , $α β + β γ + γ α = \frac{c}{a}$ , $α β γ = - \frac{d}{a}$ .

Cubic polynomial graph with roots labelled alpha, beta, gamma and Vieta's formula relationships

Common mistake

Assuming the sign pattern in Vieta's formulas is always positive — it alternates ( $- b / a$ , then $+ c / a$ , then $- d / a$ ), and the pairwise-product sign is often lost.

Mini summary

Substitute for remainders, set $P (a) = 0$ for factors, use Vieta's for root combinations without solving.

Rational Functions and Asymptotes

A rational function's long-term shape depends on comparing the degrees of numerator and denominator, while its short-term blow-ups happen wherever the denominator hits zero. A vertical asymptote occurs at $x = a$ only if that denominator factor doesn't cancel with the numerator — if it does, you get a hole instead. Horizontal asymptotes follow degree comparison, and an oblique asymptote appears when the numerator's degree is exactly one more than the denominator's.

Graph of a rational function showing vertical asymptote, horizontal asymptote, and a hole

Exam tip

Fully factorise both numerator and denominator before naming vertical asymptotes — any common factor is a hole, not an asymptote.

Mini summary

Factorise first: common factors give holes; uncancelled denominator zeros give vertical asymptotes; degree comparison gives horizontal/oblique ones.

Quick formula sheet

u_{n} = u_{1} + (n - 1) d

nth term of an arithmetic sequence — n-1 jumps, not n — the first term has zero jumps

S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})

Sum of the first n terms of an arithmetic series — Use the (u1+un) form once you know the last term

u_{n} = u_{1} r^{n - 1}

nth term of a geometric sequence

S_{n} = \frac{u _{1} ( r ^{n} - 1 )}{r - 1}, r \neq = 1

Sum of the first n terms of a geometric series — Pick the sign version that avoids dividing by a negative

S_{\infty} = \frac{u _{1}}{1 - r}, ∣ r ∣ < 1

Sum to infinity of a convergent geometric series — Always state the |r|<1 condition or lose the mark

P (a) = remainder when P (x) \div (x - a)

Remainder theorem

(x - a) ∣ P (x) ⟺ P (a) = 0

Factor theorem

α + β = - \frac{b}{a}, α β = \frac{c}{a}

Vieta's formulas for a quadratic

a x^{2} + b x + c = 0

α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β γ = - \frac{d}{a}

Vieta's formulas for a cubic

a x^{3} + b x^{2} + c x + d = 0

— Signs alternate: minus, plus, minus

Practice questions

Easy

Find the 10th term of an arithmetic sequence with $u_{1} = 4$ and $d = 3$ .
A geometric sequence has $u_{1} = 5$ and $r = 2$ . Find $u_{4}$ .
State the condition on $r$ for a geometric series to have a sum to infinity.

Medium

An arithmetic sequence has $u_{3} = 11$ and $u_{7} = 27$ . Find $u_{1}$ and $d$ .
Show that $(x - 2)$ is a factor of $P (x) = x^{3} - 3 x^{2} - 4 x + 12$ and factorise fully.
Find the equations of all asymptotes of $f (x) = \frac{x ^{2} - 1}{x - 3}$ .

Challenge

Prove by mathematical induction that $u_{n} = p + (n - 1) q$ for the recurrence $u_{1} = p$ , $u_{n + 1} = u_{n} + q$ , for all $n \in Z^{+}$ .
A ball is dropped from 10 m and rebounds to $\frac{2}{3}$ of its previous height each bounce. Find the total distance travelled until it comes to rest.
For $a x^{3} + b x^{2} + c x + d = 0$ with roots $α, β, γ$ , find $α^{2} + β^{2} + γ^{2}$ in terms of $a, b, c$ .

Frequently asked questions

What's the difference between arithmetic and geometric sequences?+

Arithmetic sequences add a constant difference $d$ each step (linear growth); geometric sequences multiply by a constant ratio $r$ each step (exponential growth or decay).

When does a geometric series have a sum to infinity?+

Only when $∣ r ∣ < 1$ , so the terms shrink towards zero. The formula is $S_{\infty} = \frac{u _{1}}{1 - r}$ , and you must state the condition to get full marks.

Is proof by induction tested at SL?+

No, induction is an HL-only technique in IB Maths AA. It's used to prove general formulas for sequences and other statements involving all positive integers.

How do I know if a rational function has a vertical asymptote or a hole?+

Fully factorise the numerator and denominator. If a factor is common to both, it cancels and creates a hole; any uncancelled denominator zero gives a vertical asymptote.

What are Vieta's formulas used for?+

They let you find combinations of a polynomial's roots (like their sum or product) directly from its coefficients, without solving the equation.

Why is the remainder theorem useful?+

It lets you find the remainder of $P (x) \div (x - a)$ by simply substituting $x = a$ into $P (x)$ , avoiding long division when you don't need the quotient.

Get the Full Number and Algebra Revision Note

Complete worked examples for every sequence, induction, and polynomial trap Full derivations for all 'show that' style questions Detailed asymptote and Vieta's formula walkthroughs with GDC tips Printable formula sheet aligned to the IB data booklet

Get the Number and Algebra notes on RevisionPrep →

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