Find the first four terms of the sequence given the following recursive formula

Formula: an = 2an-1+an-2

Given: a1 = 8, a2 = - 3

Recursive formulation:

{

a

1

=

-

4

a

n

+

1

=

5

a

n

000

(

n

=

1

,

2

,

3

,

...

)

Explanation:

We are given:

{

a

1

=

-

4

a

4

=

-

500

The general formula for the

n

th term of a geometric series is:

a

n

=

a

r

n

-

1

where

a

is the initial term and

r

is the common ratio.

A recursive formula can be given as:

{

a

1

=

a

a

n

+

1

=

r

a

n

000

(

n

=

1

,

2

,

3

,

...

)

In our example:

5

3

=

125

=

-

500

-

4

=

r

4

r

1

=

a

r

4

-

1

a

r

1

-

1

=

r

3

So the only possible Real value for

r

is

3

√

5

3

=

5

.

Footnote

There are two other possibilities for a geometric sequence with

a

1

=

-

4

and

a

4

=

-

500

, which are sequences of Complex numbers.

This is because

5

3

has two other cube roots, namely

5

ω

and

5

ω

2

, where

ω

=

-

1

2

+

√

3

2

i

is the primitive Complex cube root of

1

. Either of these will also work as a suitable common ratio.