Sequences: explicit and recursive formulas

Warm-up

Three quick questions from earlier lessons.

Discover

A sequence starts $5, 8, 11, 14, \dots$ Which rule gives the next term from the one before?

Add 3
Multiply by 3

Key idea

A sequence is a function whose inputs are the whole numbers $1, 2, 3, \dots$ : $a_n$ is its $n$ th term.

An explicit formula gives

a_n

from

n

directly (for example

a_n = 3n + 2

). A recursive formula gives a first term and a rule from one term to the next (for example

a_1 = 5

,

a_{n+1} = a_n + 3

).

Worked example

$a_1 = 2$ and $a_{n+1} = 3a_n - 1$ . Find $a_4$ .

a_2 = 3 \cdot 2 - 1 = 5

Apply the rule to a₁.

a_3 = 3 \cdot 5 - 1 = 14

Then to a₂.

a_4 = 3 \cdot 14 - 1 = 41

Then to a₃.

Your turn, guided

Your turn: $a_n = n^2 - 1$ . Find $a_5$ .

a_5 = 5^2 - 1

Replace n by 5.

a_5 = 24

25 − 1.

Practice ★

$a_n = n^2 + 5$ . Complete the table.
$a_n = n^2 + 9$ . Complete the table.

Watch out!

Practice ★★

$a_1 = 3$ and $a_{n+1} = 2a_n - 7$ . Find $a_3$ .
$a_1 = 6$ and $a_{n+1} = 2a_n - 3$ . Find $a_3$ .
$a_1 = 4$ and $a_{n+1} = 2a_n - 6$ . Find $a_3$ .

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $1,6, 11, 16, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $3,10, 17, 24, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8,14, 20, 26, \dots$

Summary card

Sequences

Explicit: $a_n$ straight from $n$ .
Recursive: a first term and a rule from $a_n$ to $a_{n+1}$ .

Watch out: check your formula with $n = 1$ .

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