Lesson 2 of 10 · 7 min
Rules that generate terms
NCERT §8.2
Meera wants the amount for week 20 without writing out every week before it. She needs a rule that turns a week number straight into an amount.
The lesson in notes
In short
Many sequences have a formula for the nth term. Even numbers 2, 4, 6, … have aₙ = 2n; odd numbers 1, 3, 5, … have aₙ = 2n − 1. For aₙ = 3n − 1 the terms are 2, 5, 8, 11, 14, … and a₂₀ = 59.
To get a particular term, substitute its position: for aₙ = (n − 2)(n + 3), a₅ = 3 × 8 = 24.
Some sequences are given by a recurrence, where each term is built from the terms before it. With a₁ = 2 and aₙ = 2aₙ₋₁ + 1 for n ≥ 2, the terms are 2, 5, 11, 23, 47, ….
The Fibonacci sequence starts a₁ = a₂ = 1 and uses aₙ = aₙ₋₂ + aₙ₋₁ for n > 2: 1, 1, 2, 3, 5, 8, 13, 21, ….
The primes 2, 3, 5, 7, 11, … have no known formula for the nth term; they can only be described in words. Not every sequence needs a formula, but every sequence needs a rule that fixes its terms in order.
A sequence is a function whose domain is the natural numbers (or a subset 1, 2, …, k of them). That is why aₙ is sometimes written a(n).