Recursive sequences
Objective. Generate and interpret sequences defined recursively.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A recursive sequence defines each term using the term(s) immediately before it, together with a starting value, rather than a direct formula in n. A common form is un+1 = f(un), such as un+1 = 2un + 1 with u1 = 3. To generate terms, substitute the previous term into the rule repeatedly: u2 = 2(3)+1 = 7, u3 = 2(7)+1 = 15, u4 = 2(15)+1 = 31. Recursive rules are useful when a pattern depends on its own history rather than its position, such as compound growth. Always state the starting term, since the same rule with a different start gives a completely different sequence.
Worked example
A sequence is defined by un+1 = 3un − 2 with u1 = 4. Find u4.
- 1u1 = 4
- 2u2 = 3(4) − 2 = 10
- 3u3 = 3(10) − 2 = 28
- 4u4 = 3(28) − 2 = 82
Answer: 82
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Pupils apply the recursive rule to the term number n instead of the previous term's value, for example computing un+1 = 2n+1 rather than 2un+1. This gives an arithmetic sequence instead of the intended one. Always substitute the actual previous term's value into the rule.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.