A conjecture is a precise statement of a rule that fits every case you have seen, written so that another case could prove it wrong. To form one, read your table for differences and for structure, write the rule with n, then test it on a case you did not use.
This follows generating systematic cases and sits inside International Mathematics investigations. The same differences technique appears in sequences and pattern rules.
How do you read a table for a rule?
Two habits help. The first is the difference method: subtract each value from the next. The second is the structure method: explain, in words or a picture, how each case is built.
Constant first differences mean the rule is linear, like 3n + 1. If the first differences themselves go up by a constant amount, the rule involves n². A structural explanation usually gives you the same rule with a reason attached.
How to move to a conjecture, step by step
- Tabulate at least five consecutive cases.
- Find first differences, and second differences if needed.
- Describe the structure of each case in a sentence.
- Write a candidate rule using n, and check that it fits every row of the table.
- State the conjecture in a full sentence, including which values of n it is for.
- Predict a new case, then build or calculate it to test the conjecture.
Worked example
In a group of n people, everyone shakes hands once with everyone else. Let H be the number of handshakes.
Step 1, table:
| n | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| H | 0 | 1 | 3 | 6 | 10 | 15 |
Step 2, differences: 1, 2, 3, 4, 5. The differences are not constant, so the rule is not linear. They go up by 1 each time.
Step 3, structure: each person shakes hands with n − 1 others. That is n × (n − 1) handshake “ends”, but each handshake has two ends and was counted twice. So H = n(n − 1)/2.
Step 4, check the table: for n = 4, 4 × 3 / 2 = 6. For n = 6, 6 × 5 / 2 = 15. Both match.
Step 5, conjecture: for every whole number n ≥ 1, the number of handshakes is n(n − 1)/2.
Step 6, new case: for n = 7 the formula gives 7 × 6 / 2 = 21. Using the difference pattern, 15 + 6 = 21. A direct count also works: 6 + 5 + 4 + 3 + 2 + 1 = 21. Three routes agree, so the conjecture has passed a real test.
The mistake to watch for
A common slip is to build a rule from just the first two or three cases.
Mistaken conjecture: from n = 1 and n = 2 (0 and 1), “H = n − 1”.
For n = 3 this gives 2, but the real count is 3.
The correction is to gather more cases before writing the rule, and to test the rule against a case that was not used to create it. Here the differences 1, 2, 3 show that the growth is speeding up, so a rule with n − 1 alone cannot be right.
Check yourself
1. Sums of the first n odd numbers are 1, 4, 9, 16, 25 for n = 1 to 5. State a conjecture and use it for n = 8.
Show answer
The values are the square numbers, so the conjecture is that the sum of the first n odd numbers is n².
For n = 8 it predicts 64. Check: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64.
2. A pattern has values 5, 8, 11, 14 for n = 1 to 4. Conjecture a rule and find the value for n = 20.
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Differences are a constant 3, so the rule starts 3n. At n = 1 we need 5, so add 2: the rule is 3n + 2.
For n = 20: 3 × 20 + 2 = 62.
3. A table gives 2, 6, 12, 20 for n = 1 to 4. Conjecture a rule and predict n = 5.
Show answer
Differences are 4, 6, 8, so they increase by 2 and the rule is quadratic. Values equal n × (n + 1): 1 × 2, 2 × 3, 3 × 4, 4 × 5.
The rule is n(n + 1), so n = 5 gives 30. Check with differences: next difference 10, and 20 + 10 = 30.
Where this leads next
Next, learn to test a counterexample to a conjecture, because a conjecture that has only met agreeing cases has not been challenged. The investigations practice set mixes all the steps.
If you often see the pattern but cannot put it into a clear sentence, a teacher can coach that wording step by step in online one-to-one Mathematics tuition. The non-calculator working trainer helps with checking each predicted value.