Testing a term rule means substituting a position that was not used to build the rule and comparing the result with what the sequence really does. It matters in any question that asks you to “show that”, “find the 50th term” or “explain why” a rule works.
This lesson follows finding a linear term rule and belongs to the module on sequences and pattern rules.
Why is one check not enough?
A rule is built from a few terms, so those terms will always fit it. Passing the terms you built it from tells you nothing new. The real test is a term further along.
For a linear rule, two agreeing terms settle it, because a straight pattern is fixed by two points. For anything curved, you need a term beyond the ones you used, and you should say what you assumed.
How to test a rule, step by step
- Pick a distant position not used in building the rule, such as n = 10 or n = 25.
- Substitute that position into your rule.
- Find the same term another way, by counting on in steps or by continuing the pattern.
- Compare. If the two values agree, the rule passes. If they differ, find the error before going further.
- State any assumption if the question gave only a few terms.
Worked example
A student says the nth term of 7, 10, 13, 16, … is 3n + 4. Test the rule and find the 12th term.
Step 1, distant position: choose n = 12.
Step 2, substitute: 3 × 12 + 4 = 36 + 4 = 40.
Step 3, count on: the first term is 7 and there are 11 steps of 3, so 7 + 11 × 3 = 7 + 33 = 40.
Step 4, compare: both routes give 40, so the rule passes. The 12th term is 40.
Notice that the two routes are independent. The rule uses the formula, and the counting uses only the original pattern.
The mistake to watch for
The more serious slip is to treat a short list as proof.
Mistaken answer: “The terms 1, 2, 4 double each time, so the nth term is 2^(n − 1) and the 10th term is 512.”
The student found one pattern that fits and never checked whether another does.
A quadratic rule, (n² − n + 2)/2, also gives 1, 2, 4 for n = 1, 2, 3. At n = 4 it gives (16 − 4 + 2)/2 = 7, while doubling gives 8.
Both rules are valid for the three given terms, and only a fourth term can separate them. The correction is to state the assumption (“assuming the sequence keeps doubling”) or to use the extra information in the question.
Check yourself
Try these without a calculator, then open each answer.
1. The sequence 7, 10, 13, 16, … has rule 3n + 4. Test the rule at n = 20.
Show answer
Formula: 3 × 20 + 4 = 64. Counting on: 7 + 19 × 3 = 7 + 57 = 64. The values agree, so the rule passes and the 20th term is 64.
2. A student says the nth term of 5, 8, 11, … is 2n + 3. Show with one term that the rule is wrong, then give the correct rule.
Show answer
At n = 3 the rule gives 2 × 3 + 3 = 9, but the third term is 11. So the rule fails.
The difference is 3, so the rule starts with 3n. Since 3n gives 3, 6, 9 and the terms are 5, 8, 11, the constant is 2. The correct rule is 3n + 2. Check at n = 3: 9 + 2 = 11.
3. The terms 2, 4, 8 fit the rule “double each time”. They also fit n² − n + 2. Find the 4th term from each rule.
Show answer
Doubling: 8 × 2 = 16. Quadratic: 4² − 4 + 2 = 16 − 4 + 2 = 14.
The rules agree on three terms and disagree on the fourth, so three terms alone cannot decide between them.
Where this leads next
The next question is what kind of change a sequence follows, which is the focus of distinguishing arithmetic from geometric change. After that, the sequences practice set mixes all the skills. The sequence and series laboratory lets you generate terms from two rules and see where they split.
Some students find the testing step feels like extra work until the first time it catches a wrong rule. A teacher in online one-to-one Mathematics tuition can help you make it a habit that takes seconds.