contestada

For the following geometric sequence find the recursive formula and the 5th term in the sequence. In your final answer, include all of your work.
{-4, 12, -36, ...}

Respuesta :

calculista
we have a geometric sequence----------- > {-4, 12, -36, ...}

the formula is a(r)^(n-1)

a------------- >a is the first term------------ > -4
r--------------- > is the common ratio------- > 12/(-4)=(-36/12)=-3
n--------------- > is the number of terms

The fifth term is -4[(-3)^(5-1)]=-4[(81]=-324

the answer is -324
AlpenGlow
First you need to identify the common ratio:

r = [tex] t_{n+1} / t_{n} [/tex]

where r is the common ratio
          [tex] t_{n} [/tex] is any term in the sequence
          [tex] t_{n} [/tex] is the term preceding [tex] t_{n} [/tex]

In your case the common ratio will be 
  12/-4 = -3
The recursive formula of a geometric sequence is:

An = [tex] A_{n-1} [/tex] x r
Where : An is the nth term
             [tex] A_{n-1} [/tex] is the erm preceding the nth term
             r = common ratio

For this case it is:
An = [tex] A_{n-1} [/tex] x -3

Now let us use this formula to find the 5th term:

[tex] A_{5} [/tex] = [tex] A_{5-1}[/tex] x -3
[tex] A_{5} [/tex] = [tex] A_{4}[/tex] x -3

Since you do not know the 4th term, you can use that by using our reclusive formula:

[tex] A_{4} [/tex] = [tex] A_{4-1} [/tex] x -3
[tex] A_{4} [/tex] = [tex] A_{3} [/tex] x -3
[tex] A_{4} [/tex] = [tex] -36 [/tex] x -3
[tex] A_{4} [/tex] = [tex] 108 [/tex]

Now that you know your fourth term, you can use the same formula:

[tex] A_{5} [/tex] = [tex] A_{5-1} [/tex] x -3
[tex] A_{5} [/tex] = [tex] A_{4} [/tex] x -3
[tex] A_{5} [/tex] = [tex] 108 [/tex] x -3
[tex] A_{5} [/tex] = [tex] -324 [/tex] 

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