The sequence and series laboratory works in two ways. In the first, you give it a, d or r, and n, and it shows the terms, the nth term and the sum with every step written out. In the second, you give it a list of terms and it tests which simple rules could produce them.
Its main lesson is that a few terms can fit more than one pattern, so a rule has to be justified, not guessed.
How do you use it?
If you know a, d or r, and n:
- Choose I know a, d or r, and n.
- Choose Arithmetic (common difference d) or Geometric (common ratio r).
- Enter the first term a, the common difference d or ratio r, and the number of terms n.
- Press Show working.
If you have a list:
- Choose I have a list of terms.
- Type the terms in order, separated by commas, for example 1, 2, 4.
- Press Show working.
Reset returns the sample.
How do you read the result?
For the first mode you see the first terms, the nth term and the sum of the first n terms, each with the formula, the substitution and the answer. For a geometric sequence the tool also says whether an infinite sum exists, based on whether |r| is below 1.
For the list mode you see the first differences and the ratios of consecutive terms, then a list of candidate rules: arithmetic, geometric and quadratic. Each candidate shows its next term. If more than one rule fits, a warning explains that finite data does not give one unique rule.
Example walk-through
The sample is arithmetic with a = 3, d = 2 and n = 5. The terms are 3, 5, 7, 9, 11. The nth term is 3 + (5 − 1) × 2 = 11.
The sum is 5 ÷ 2 × (2 × 3 + 4 × 2) = 5 ÷ 2 × 14 = 35. You can check by adding: 3 + 5 + 7 + 9 + 11 = 35.
Switch to geometric with a = 3, r = 2 and n = 5. The terms are 3, 6, 12, 24, 48, and the sum is 3(1 − 2⁵) ÷ (1 − 2) = 93. Since |r| is 2, there is no infinite sum. Use r = 0.5 instead and the tool gives 3 ÷ (1 − 0.5) = 6.
Now switch to the list mode and enter 1, 2, 4. The differences are 1 and 2, so it is not arithmetic. The ratios are both 2, so it fits a geometric rule and the next term would be 8.
The second difference is constant at 1, so it also fits a quadratic rule, and the next term would be 7. Two rules, two different next terms. That is the point of the lab.
What are the assumptions and limits?
- The parameter mode assumes the sequence is exactly arithmetic or geometric from the first term, and that n is a positive whole number (up to 1000).
- The list mode accepts 2 to 30 terms.
- Finding a rule that fits does not make it the rule. It stays a conjecture until you prove it in general.
- The tool tests arithmetic, geometric and quadratic rules only. Other patterns exist and will not be found.
- An infinite geometric sum needs |r| < 1. The tool will say when it does not exist.
Which lessons explain the ideas behind it?
- Sequences and pattern rules practice gives mixed questions on spotting and justifying rules.
- Find a term from two sequence conditions covers using two facts to find a and d or r.
- Calculate a finite arithmetic sum explains the sum formula used here.
- Recover a common ratio from terms works backwards from given terms.
- Use an infinite sum only when its condition holds explains the |r| < 1 check.
- Arithmetic and geometric series practice gives mixed questions.
The wider topic is arithmetic and geometric series. For a teacher to check your reasoning on real questions, see online one-to-one Additional Mathematics tuition. Other tools are in the learning tools directory.