A recursive rule says how to get the next term from the previous one, and states where to start.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The starting value is essential
aₙ = aₙ₋₁ + 5 describes infinitely many sequences until a₁ is given.
Explicit rules
An explicit rule computes any term directly from its position, with no earlier terms required.
Which is better
Recursive rules describe the pattern most naturally. Explicit rules answer "what is the 500th term" without computing 499 others.
Converting
A recursive rule adding d becomes a₁ + (n − 1)d. One multiplying by r becomes a₁ · r^(n−1).
Not everything converts
The Fibonacci sequence adds the two previous terms. Its recursive rule is simple and its explicit one is not.
Recursive rules build from the previous term
aₙ = aₙ₋₁ + d describes how the sequence is generated, and needs a stated first term. It mirrors how a sequence actually unfolds, which makes it the more natural description.
Explicit rules jump straight there
To find the hundredth term from a recursive rule you must compute the ninety-nine before it. An explicit rule gives it directly, which is why both forms are kept.
Converting between them
From a recursive rule, identify the first term and the operation, then write the matching explicit form. Going the other way, read the starting value and the step out of the formula.
Recursion describes more than sequences
Some sequences — the Fibonacci numbers, for instance — have a simple recursive rule and an awkward explicit one. Recursion is the more general tool, which is why computing uses it so heavily.
Step 2: Try It Yourself
Tap and try it out.
The slope is 3: for every 1 across, the line goes 3 up.
Step 3: Watch an Example
One step at a time.
Watch Sana Convert a Rule
A sequence is defined by a₁ = 7 and aₙ = aₙ₋₁ + 5.
- Step 1
The rule adds 5 each time, so this is arithmetic with d = 5.
Step 4: Your Turn
Practice makes it stick.
The Recursion
Problem 1 of 2
a₁ = 7 and aₙ = aₙ₋₁ + 5. What is the 3rd term?
The Explicit Rule
Problem 2 of 2
aₙ = 7 + 5(n − 1). What is the 10th term?
Two Kinds of Rule
1 of 8
a₁ = 2 and aₙ = aₙ₋₁ + 6. What is the 4th term?
2 of 8
a₁ = 3 and aₙ = 2aₙ₋₁. What is the 4th term?
3 of 8
aₙ = 4 + 3(n − 1). What is the 6th term?
4 of 8
aₙ = 5 · 2^(n−1). What is the 4th term?
5 of 8
Which rule finds the 500th term without the earlier ones? 1 recursive, 2 explicit.
6 of 8
Fibonacci: 1, 1, 2, 3, 5. What is the next term?
7 of 8
Match each rule with what it needs to compute a term.
Tap a card on the left to start.
8 of 8
aₙ = 10 + 4(n − 1). What is the 8th term?
Step 5: Quick Check
Show what you know.
Question 1 of 2
aₙ = 6 + 5(n − 1). What is the 7th term?
Question 2 of 2
Why must a recursive rule state a starting value?
What You Learned
- A recursive rule builds each term from the previous one and needs a starting value.
- An explicit rule computes any term straight from its position.
- Arithmetic and geometric sequences convert easily between the two.