# Recursive Formula

How to solve recursive formula problems: examples and their solutions.

## Example 1

The recursive formula is a way
to express a sequence.

There are two parts:

Initial term(s): a1 (or a2 if needed)
Difference equation (an and an + 1)

To find a2,

put a1 = 4
into a2 = a1 + 6.

a2 = 4 + 6
= 10

To find a3,

put a2 = 10
into a3 = a2 + 6.

a3 = 10 + 6
= 16

To find a4,

put a3 = 16
into a4 = a3 + 6.

a4 = 16 + 6
= 22

So the first four terms are 4, 10, 16, and 22.

## Example 2

To find a2,

put a1 = 3
into a2 = 2a1.

a2 = 2⋅3
= 6

To find a3,

put a2 = 6
into a3 = 2a2.

a3 = 2⋅6
= 12

To find a4,

put a3 = 12
into a4 = 2a3.

a4 = 2⋅12
= 24

So the first four terms are 3, 6, 12, and 24.

## Example 3

To find a2,

put a1 = -1
into a2 = a1 + 3⋅1.

a2 = -1 + 3⋅1
= 2

To find a3,

put a2 = 2
into a3 = a2 + 3⋅2.

a3 = 2 + 3⋅2
= 8

To find a4,

put a3 = 8
into a4 = a3 + 3⋅3.

a4 = 8 + 3⋅3
= 17

So the first four terms are -1, 2, 8, and 17.

## Example 4

To find a3,

put a1 = 1 and a1 = 1
into a3 = a1 + a2.

a3 = 1 + 1
= 2

To find a4,

put a2 = 1 and a3 = 2
into a4 = a2 + a3.

a4 = 1 + 2
= 3

To find a5,

put a3 = 2 and a4 = 3
into a5 = a3 + a4.

a5 = 2 + 3
= 5

To find a6,

put a4 = 3 and a5 = 5
into a6 = a4 + a5.

a6 = 3 + 5
= 8

So the first six terms are 1, 1, 2, 3, 5, and 8.

## Example 5

The difference between an + 1 and an is constant: 6.

So this recursive formula
shows an arithmetic sequence.

Write the difference equations vertically,
starting from a2 to an.

a2 = a1 + 6
a3 = a2 + 6
a4 = a3 + 6
...
an = an - 1 + 6

The gray terms on both sides are cancelled.
(a2 ~ an - 1)
And '+6' is added 'n - 1' times.

So an = a1 + (n - 1)⋅6.

Arrange the terms.

Then an = 6n - 2.

## Example 6

The ratio between an + 1 and an is constant: 2.

So this recursive formula
shows a geometric sequence.

Write the difference equations vertically,
starting from a2 to an.

a2 = 2a1
a3 = 2a2
a4 = 2a3
...
an = 2an - 1

Then multiply these equations.

The gray terms on both sides are cancelled.
(a2 ~ an - 1)
And '2' is multiplied 'n - 1' times.

So an = a1⋅2n - 1.

Arrange the terms.

Then an = 3⋅2n - 1.