To state the limits of a conclusion, say which cases you tested, whether you proved the rule, and which values it is claimed for, and claim nothing beyond that. Precise wording turns a pattern you found into a mathematical statement other people can trust.
This is the final step of the route in International Mathematics investigations. It follows explaining a rule with algebra and links back to testing counterexamples.
What are the three levels of a conclusion?
The strength of your claim depends on your evidence. There are three levels worth naming.
| Level | What you did | How to word it |
|---|---|---|
| Observation | Checked some cases | “The rule fits n = 1 to 5.” |
| Conjecture | Tested on new cases as well | “I conjecture the rule holds for n ≥ 1.” |
| Proved | Algebra or a full argument | “This is true for every n ≥ 1.” |
Only the last level allows “always” or “for all”. The first two need the word “conjecture”, “appears to” or a stated range.
How to state limits, step by step
- List what you tested, for example n = 1 to 6.
- Say if the rule was proved and how, or say it is a conjecture.
- State the domain: the values of n for which the rule is claimed.
- Check the boundary, usually the smallest case, and mention any exception.
- Label predictions beyond your evidence as predictions.
- Note the conditions in the situation, such as points in general position or a physical maximum.
Worked example
Points are marked on a circle and every pair is joined by a straight chord. The chords cut the circle into regions.
The points are placed so that no three chords meet at a single point. Let R be the number of regions for n points.
Step 1, table:
| n | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| R | 1 | 2 | 4 | 8 | 16 | 31 |
Step 2, observation: for n = 1 to 5 the values double each time, which fits R = 2n−1. For n = 6 the rule predicts 32.
Step 3, compare with evidence: a careful count for n = 6 gives 31, not 32. So R = 2n−1 fails at n = 6. By the counterexample idea, the conjecture is false.
Step 4, honest conclusion: “R = 2n−1 fits n = 1 to 5 only. It does not hold for n = 6, so it is not a general rule for the number of regions.”
The lesson for your own investigations: five agreeing cases were not enough. Also the condition about no three chords meeting matters, because some arrangements of six points give fewer regions, so the claim is about points placed in general position.
The mistake to watch for
A common slip is to write a conclusion that goes beyond the evidence.
Mistaken conclusion: “The number of regions is always 2n−1, as shown by n = 1 to 5.”
“Always” was never earned, and n = 6 contradicts it.
The correction is to match the wording to the evidence. For a rule that fits cases only, write “appears to hold for n = 1 to 5”. For a rule with a proof, write “holds for all n ≥ 1, because…” and give the reason.
Check yourself
1. A square n by n is made of tiles, and only its border tiles are coloured. The counts for n = 2 to 6 are 4, 8, 12, 16, 20. A student writes 4n − 4. What limit should the student state, given that n = 1 is a single tile?
Show answer
For n = 1 the rule gives 4 − 4 = 0, but the single tile is a border tile, so the real count is 1. The rule is valid only for n ≥ 2.
For n = 10 it gives 36, which agrees with 100 − 64 = 36 (total tiles minus the 8 × 8 inner square).
2. A student tests a rule for n = 1, 2, 3 and 4, and writes “so it is true for all n”. Rewrite the conclusion honestly.
Show answer
For example: “The rule fits n = 1 to 4. I conjecture that it holds for all n ≥ 1 but I have not proved it.” This states the evidence, the claim and the status.
3. The sum of the first n odd numbers fits n² for n = 1 to 5. Use it to predict the sum for n = 30, and say how reliable that prediction is.
Show answer
The prediction is 30² = 900.
From the pattern alone it is only a prediction. It becomes certain if you prove the rule, for example with the sum of an arithmetic series: n/2 × (1 + (2n − 1)) = n².
Where this leads next
Put all five steps together in the investigations practice set. If you want to keep a record of limits you forgot to state, the mistake log and retest queue is a helpful companion, and the non-calculator working trainer supports quick value checks.
A teacher can read your written conclusions and show where the wording claims too much or too little. That kind of feedback is part of online one-to-one Mathematics tuition.