Recursive Formula

Recursive Formula

How to find the terms of a sequence in recursive formula: examples and their solutions.

Example 1

Find the first four terms of the given sequence. a1 = 4, a(n + 1) = an + 6

The recursive formula is a way
to express a sequence.

There are two parts:

Initial term(s): a1 (or a2 if needed)
Equation with an and an + 1

an + 1 = an + 6

So a2 = a1 + 6.

a1 = 4

So a2 = 4 + 6.

4 + 6 = 10

So a2 = 10.

an + 1 = an + 6

So a3 = a2 + 6.

a2 = 10

So a3 = 10 + 6.

10 + 6 = 16

So a3 = 16.

an + 1 = an + 6

So a4 = a3 + 6.

a3 = 16

So a4 = 16 + 6.

16 + 6 = 22

So a4 = 22.

a1 = 6
a2 = 10
a3 = 16
a4 = 22

So the first four terms are
6, 10, 16, and 22.

Example 2

Find the first four terms of the given sequence. a1 = 3, a(n + 1) = 2*an

an + 1 = an + 3n

So a2 = a1 + 3⋅1.

a1 = -1
+3⋅1 = +3

So a2 = -1 + 3.

-1 + 3 = 2

So a2 = 2.

an + 1 = an + 3n

So a3 = a2 + 3⋅2.

a2 = 2
+3⋅2 = +6

So a3 = 2 + 6.

2 + 6 = 8

So a3 = 8.

an + 1 = an + 3n

So a4 = a3 + 3⋅3.

a3 = 8
+3⋅3 = +9

So a4 = 8 + 9.

8 + 9 = 17

So a4 = 17.

a1 = -1
a2 = 2
a3 = 8
a4 = 17

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

Example 3

Find the first four terms of the given sequence. a1 = 1, a2 = 1, a(n + 2) = an + a(n + 1)

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

This sequence is called the [Fibonacci Sequence].

an + 2 = an + an + 1

So a3 = a1 + a2.

a1 = 1
a2 = 1

So a3 = 1 + 1.

1 + 1 = 2

So a3 = 2.

an + 2 = an + an + 1

So a4 = a2 + a3.

a2 = 1
a3 = 2

So a4 = 1 + 2.

1 + 2 = 3

So a4 = 3.

an + 2 = an + an + 1

So a5 = a3 + a4.

a3 = 2
a4 = 3

So a5 = 2 + 3.

2 + 3 = 5

So a5 = 5.

an + 2 = an + an + 1

So a6 = a4 + a5.

a4 = 3
a5 = 5

So a6 = 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, and 8.

Example 4: Arithmetic Sequence in Recursive Formula

For the given sequence, find an. a1 = 4, a(n + 1) = an + 6

See the given recursive formula:
an + 1 = an + 6.

It means
to find the next term,
add the last term and +6.

So this is an arithmetic sequence
in recursive formula:
an + 1 = an + d.

Arithmetic sequence

You can directly use [a1 = 4] and [d = 6]
to write an.

But, let's write an
without directly using the formula.

an + 1 = an + 6

So a2 = a1 + 6.

By the same way,
a3 = a2 + 6,
a4 = a3 + 6,
....

And write
an = an - 1 + 6.

Add these equations.

The gray terms will be cancelled.

The left side is an.

In the right side,
a1 remains.
And (n - 1) of 6 remain.

So the right side is
a1 + (n - 1)⋅6.

a1 = 4

So an = 4 + (n - 1)⋅6.

As you can see,
this is the an formula of the arithmetic sequence.
(a1 = 4, d = 6)

(n - 1)⋅6 = 6n - 6

4 - 6 = -2

So an = 6n - 2.

Example 5: Geometric Sequence in Recursive Formula

For the given sequence, find an. a1 = 3, a(n + 1) = 2an

See the given recursive formula:
an + 1 = 2an.

It means
to find the next term,
multiply 2 and the last term.

So this is a geometric sequence
in recursive formula:
an + 1 = ran.

Geometric sequence

You can directly use [a1 = 3] and [r = 2]
to write an.

But, let's write an
without directly using the formula.

an + 1 = 2an

So a2 = 2⋅a1.

By the same way,
a3 = 2⋅a2,
a4 = 2⋅a3,
....

And write
an = 2⋅an - 1.

Multiply these equations.

The gray terms will be cancelled.

The left side is an.

In the right side,
(n - 1) of 2 remain.
and a1 remains.

So the right side is
2n - 1a1.

a1 = 3

So an = 2n - 1⋅3.

Switch the order of 2n - 1 and 3.

Then an = 3⋅2n - 1.

As you can see,
this is the an formula of the geometric sequence.
(a1 = 3, r = 2)