Build it from the last one

The idea

A recursive rule describes a sequence by giving its first term and relating each term tₙ to previous terms such as tₙ₋₁; the earlier terms must be known to find the next ones.

Only the last step

There is another way to describe a sequence. Take 1, 4, 7, 10, 13, … Each term is 3 more than the term before it.

You know only this: the first term is 1, and each term is 3 more than the one before. How would you find the 10th term?

• Add 3 again and again from the start
• Jump straight to it
• Multiply 10 by 3

One term at a time

Press the button to find each next term. Try tapping the locked t₁₀ too.

The recursive rule

A recursive rule describes a sequence by giving its first term and relating each term tₙ to previous terms such as tₙ₋₁. The earlier terms must be known to find the next ones.

For 1, 4, 7, 10, 13, … : t₁ = 1 and tₙ = tₙ₋₁ + 3 for n ≥ 2. So t₂ = t₁ + 3, t₃ = t₂ + 3, t₄ = t₃ + 3, and so on.

The same sequence also has the explicit rule tₙ = 3n − 2. One sequence, two ways to describe it.

Notes

A recursive rule gives the first term and relates each term tₙ to previous terms such as tₙ₋₁; the earlier terms must be known to find the next ones.

Check yourself

A sequence has u₁ = 1 and uₙ = 2uₙ₋₁ + 3 for n ≥ 2. What is u₃?

Answer: 13

u₂ = 2 × 1 + 3 = 5. Then u₃ = 2 × 5 + 3 = 10 + 3 = 13.

A sequence has t₁ = 2 and tₙ = tₙ₋₁ + 5 for n ≥ 2. What is t₅?

Answer: 22

t₂ = 7, t₃ = 12, t₄ = 17, t₅ = 22.

To use a recursive rule to find t₈, what must you know?

A sequence has s₁ = 3 and sₙ = sₙ₋₁ × (sₙ₋₁ − 1) for n ≥ 2. What is s₃?

Answer: 30

s₂ = 3 × (3 − 1) = 3 × 2 = 6. Then s₃ = 6 × (6 − 1) = 6 × 5 = 30.

• Only the number 8. That is how an explicit rule works. A recursive rule builds a term from the terms before it.
• Nothing: the rule is enough. The rule says how to get a term from the previous one, so you need the previous term too.
• The term before it, t₇ — correct. Yes. The earlier terms must be known to find the next ones.