Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Finding formulas for 1ᵏ + 2ᵏ + … + nᵏ

AA HLAA SL Solid

Research question

Can I derive formulas for the sums of squares, cubes and fourth powers using finite differences and a telescoping identity, and what pattern links their coefficients?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

The formula for Σk² is quoted in many textbooks but rarely derived. Finding it yourself — and the next few — by two different methods, then proving them by induction, is pure mathematical exploration.

The mathematics you'll need

Sigma notation and series
Method of differences and telescoping sums
Solving simultaneous equations for polynomial coefficients
Proof by induction (HL)
Patterns in coefficients (Bernoulli numbers — explain)

Where the data comes from

No data needed.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

A possible outline

Conjecture the degree of each sum.
Find coefficients by fitting polynomials.
Derive with the telescoping identity (k+1)³ − k³.
Prove by induction.
Look for patterns and reflect on generalising.

Pitfalls that cost marks

Only one method.
Proofs with gaps in the inductive step.
Introducing Bernoulli numbers without understanding them.

Showing personal engagement

Discover a pattern (e.g., Σk³ = (Σk)²) and prove it.
Explain Gauss's school story and extend it.
Visualise Σk³ = (Σk)² geometrically.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Sigma notation and series; Method of differences and telescoping sums
AA HL	Good fit	Sigma notation and series; Method of differences and telescoping sums
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: only one method; and proofs with gaps in the inductive step. Say how each one affects your answer.

Use of mathematics (E)

SL: sigma notation and series; method of differences and telescoping sums; solving simultaneous equations for polynomial coefficients; patterns in coefficients (Bernoulli numbers — explain), each explained in your own words and carried out correctly.

HL adds: proof by induction, each explained in your own words and carried out correctly.

How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Derive the sum of fifth powers and compare with the general Faulhaber formula.

Extending it for HL

This idea already has HL mathematics in it: Proof by induction. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

How far can a stack of books lean over the edge of a table? (AA HL) is a complete pure maths exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Sigma notation and series, method of differences and telescoping sums, solving simultaneous equations for polynomial coefficients, proof by induction (HL) and patterns in coefficients (Bernoulli numbers — explain).

Where can I get data for this IA?

No data needed. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).

Turn this idea into your IA

The planner opens with “Finding formulas for 1ᵏ + 2ᵏ + … + nᵏ” filled in: the question, the maths and the data sources.

Plan this IA free

Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in

Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)

Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · annotated exemplars, marked on every criterion