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?

Adapt it: change the place, the data or the comparison until the question is yours.

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)

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

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

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).