Explicit and recursive rules · 2 marks

Consider the expression u_n = 2n − 1. This states that the nth term of the sequence is given by the rule 2n − 1.

Answer: u₁ = 1, u₂ = 3, u₃ = 5, … so u_n = 2n − 1 is the explicit rule for the sequence of odd numbers 1, 3, 5, 7, ….

Step-by-step solution

Idea: An explicit rule gives a term straight from its position n. Put n = 1, 2, 3, … into 2n − 1 to see which sequence it makes.

n = 1: u₁ = 2 × 1 − 1 = 1.½ mark
n = 2: u₂ = 2 × 2 − 1 = 3; n = 3: u₃ = 2 × 3 − 1 = 5.½ mark
Similarly u₄ = 7, u₅ = 9, … The terms are 1, 3, 5, 7, 9, … Why odd? 2n is always even, and one less than an even number is odd.½ mark
So u_n = 2n − 1 is the explicit rule for the nth term of the sequence of odd numbers.½ mark

Substituting n = 1, 2, 3, … gives 1, 3, 5, …: uₙ = 2n − 1 is the explicit rule for the odd numbers.

Check: The 10th odd number is 19, and 2 × 10 − 1 = 19 ✓.

u_n = 2n − 1

u₁ = 2 × 1 − 1 = 1

u₂ = 2 × 2 − 1 = 3

u₃ = 2 × 3 − 1 = 5

∴ 1, 3, 5, 7, … : u_n = 2n − 1 is the explicit rule for the odd numbers.

Calculating 2n − 1 as 2(n − 1). For n = 3, 2 × 3 − 1 = 5, not 2 × 2 = 4: multiply first, then subtract.
Starting with n = 0 and getting −1. Positions start at n = 1.
Writing the terms as 1, 2, 3, … (the positions) instead of the values 1, 3, 5, ….

MCQ (1 mark)

The explicit rule u_n = 2n − 1 gives the 25th odd number as

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(B) 49
u₂₅ = 2 × 25 − 1 = 49.

Write the first five terms of the sequence u_n = 2n + 1. How is it related to the odd numbers?

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3, 5, 7, 9, 11. These are the odd numbers starting from 3 (the odd numbers without the first one, 1).