Recursive Formula

How to find the first few terms of the given sequence in recursive formula: 3 examples and their solutions.

Example 1

Example

The recursive formula is a way
to write a sequence.

There are two parts:
Initial condition (a1),
Equation with an and its adjacent term(s).

Solution

a1 is given.

an + 1 = an + 6

Then a2 = a1 + 6.

a1 = 4

So a1 + 6 = 4 + 6.

4 + 6 = 10

So a2 = 10.

You found a2.

an + 1 = an + 6

Then a3 = a2 + 6.

a2 = 10

So a2 + 6 = 10 + 6.

10 + 6 = 16

So a3 = 16.

You found a3.

an + 1 = an + 6

Then a4 = a3 + 6.

a3 = 16

So a3 + 6 = 16 + 6.

16 + 6 = 22

So a4 = 22.

a1 = 4
a2 = 10
a3 = 16
a4 = 22

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

So
4, 10, 16, 22
is the answer.

Example 2

Example

Solution

a1 is given.

an + 1 = an + 3n

Then a2 = a1 + 3⋅1.

a1 = -2

So a1 + 3⋅1 = -2 + 3⋅1.

+3⋅1 = +3

-2 + 3 = 1

So a2 = 1.

You found a2.

an + 1 = an + 3n

Then a3 = a2 + 3⋅2.

a2 = 1

So a2 + 3⋅2 = 1 + 3⋅2.

+3⋅2 = +6

1 + 6 = 7

So a3 = 7.

You found a3.

an + 1 = an + 3n

Then a4 = a3 + 3⋅3.

a3 = 7

So a3 + 3⋅3 = 7 + 3⋅3.

+3⋅3 = +9

7 + 9 = 16

So a4 = 16.

a1 = -2
a2 = 1
a3 = 7
a4 = 16

So the first four terms are
-2, 1, 7, 16.

So
-2, 1, 7, 16
is the answer.

Example 3

Example

an + 2 = an + an + 1 means
to find the next term,
add the last two terms.

This sequence is the Fibonacci Sequence.

Solution

a1 and a2 are given.

an + 2 = an + an + 1

Then a3 = a1 + a2.

a1 = 1
a2 = 1

So a1 + a2 = 1 + 1.

1 + 1 = 2

So a3 = 2.

a2 is given.
And you found a3.

an + 2 = an + an + 1

Then a4 = a2 + a3.

a2 = 1
a3 = 2

So a2 + a3 = 1 + 2.

1 + 2 = 3

So a4 = 3.

You found a3 and a4.

an + 2 = an + an + 1

Then a5 = a3 + a4.

a3 = 2
a4 = 3

So a3 + a4 = 2 + 3.

2 + 3 = 5

So a5 = 5.

You found a4 and a5.

an + 2 = an + an + 1

Then a6 = a4 + a5.

a4 = 3
a5 = 5

So a4 + a5 = 3 + 5.

3 + 5 = 8

So a6 = 8.

a1 = 1
a2 = 1
a3 = 2
a4 = 3
a5 = 5
a6 = 8

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

So
1, 1, 2, 3, 5, 8
is the answer.