The position n tells you where a term sits in the list. The value of the term is the number found there. A rule takes a position in and gives a value out, and questions can ask either way round.
This lesson rounds off the module on sequences and pattern rules, using the rules you learnt in finding a linear term rule.
Why do these two ideas get mixed up?
In the sequence 3, 8, 13, 18, the number 8 is the second term. People say “8” and “2” in the same breath.
The exam wording does not always help. “Find the 20th term” gives a position and asks for a value. “Which term is 98?” gives a value and asks for a position.
The rule 5n − 2 works in one direction directly: put n in, get the value out. The other direction needs an equation.
How to work through it, step by step
- Read the question: is it giving you a position or a value?
- Position given: substitute it for n.
- Value given: set the rule equal to the value and solve for n.
- Check n: it must be a positive whole number. If it is not, the value is not in the sequence.
- Answer in the question’s terms: say “the 20th term” for a position, or give the number for a value.
Worked example
The nth term of a sequence is 5n − 2, giving 3, 8, 13, 18, …
(a) Which term equals 98? Set 5n − 2 = 98. Then 5n = 100 and n = 20. The value 98 is the 20th term.
(b) Is 150 a term? Set 5n − 2 = 150. Then 5n = 152 and n = 30.4. This is not a whole number, so 150 is not a term.
(c) What is the first term greater than 200? Solve 5n − 2 > 200. Then 5n > 202 and n > 40.4. The smallest whole position is 41. The value is 5 × 41 − 2 = 203. Check: position 40 gives 198, which is below 200.
Each answer uses n only to find the position and only reports a value when the question asks for one.
The mistake to watch for
The common slip is to put a value where n belongs.
Mistaken answer: “Which term is 98?” 5 × 98 − 2 = 488, so it is the 488th term.
The student treated 98 as the position and substituted it for n.
The question gave a value, so 98 must equal the whole rule, not n. The test is to ask which side of the rule the number belongs on. Positions go into the rule, and values come out of it, so a given value sits on the other side of the equals sign.
Check yourself
Try these without a calculator, then open each answer.
1. The nth term is 4n + 3. Which term equals 83?
Show answer
Set 4n + 3 = 83. Then 4n = 80 and n = 20. The value 83 is the 20th term.
Check: 4 × 20 + 3 = 83.
2. Is 100 a term of the sequence with nth term 4n + 3?
Show answer
Set 4n + 3 = 100. Then 4n = 97 and n = 24.25. This is not a whole number, so 100 is not a term.
3. Find the first term of the sequence 4n + 3 that is greater than 500.
Show answer
Solve 4n + 3 > 500. Then 4n > 497 and n > 124.25, so the smallest position is 125. The value is 4 × 125 + 3 = 503.
Check: position 124 gives 499, which is below 500.
Where this leads next
Put the whole module together with the sequences practice set. The sequence and series laboratory lets you try a rule and see which positions give a chosen value. The non-calculator working trainer helps with the equation steps.
Questions that depend on reading the wording correctly are where a teacher in online one-to-one Mathematics tuition can be most useful, because the slip is in the interpretation and not in the calculation.