Jump to any term

The idea

An explicit rule gives the nth term tₙ as a formula in the position number n, so any term can be found directly by substituting n, without knowing the previous terms.

The 20th odd number

The odd numbers are 1, 3, 5, 7, 9, 11, … Writing them all out to find the 20th odd number is slow.

Is there a way to find the 20th odd number without writing the first 19?

• Yes, a rule can give it directly
• No, I have to write them all

Try the machine

This machine uses the rule uₙ = 2n − 1. Put in a position number n and read the term. No earlier terms needed.

The explicit rule

An explicit rule gives the nth term tₙ as a formula in the position number n. Substitute n and you get any term directly, without knowing the previous terms.

For uₙ = 2n − 1 we get u₁ = 2 × 1 − 1 = 1, u₂ = 2 × 2 − 1 = 3, u₃ = 2 × 3 − 1 = 5, and so on. These are the odd numbers.

Notes

An explicit rule gives the nth term tₙ as a formula in the position number n, so any term can be found directly by substituting n, without knowing the previous terms.

Check yourself

For uₙ = 2n − 1, what is u₁₀?

Answer: 19

u₁₀ = 2 × 10 − 1 = 20 − 1 = 19.

The rule sₙ = 5n − 2 gives a sequence. What is the 100th term?

Answer: 498

s₁₀₀ = 5 × 100 − 2 = 500 − 2 = 498.

To find the 53rd term of uₙ = 2n − 1, what do you need?

The sequence 1, 4, 7, 10, 13, … has the explicit rule tₙ = 3n − 2. What is t₁₀?

Answer: 28

t₁₀ = 3 × 10 − 2 = 30 − 2 = 28.

• The position number n = 53 only — correct. Yes. With an explicit rule you substitute n = 53 and get u₅₃ = 105 directly.
• All the 52 terms before it. No need. An explicit rule gives a term directly from its position number.
• The 52nd term. An explicit rule does not use the term before. It uses only the position number n.