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30 December, 12:57

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

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

Given: a1 = 8, a2 = - 3

+3
Answers (1)
  1. 30 December, 15:00
    0
    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.
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