find the next three terms in the geometric sequence, 800, 200, 50, 25/2, ...

Question

Find the next three terms in the geometric sequence. 800, 200, 50, 25/2, ...

Answer 1

Since we are told this is a geometric sequence, we know that it has the form (by definition): 

 

a, ar, ar², ar³,...  where r is an arbitrary value,

 

Where here a = 800.

 

To get the value of r, divide any two terms:  For example, 200/800 = 1/4.   So r = 1/4.  Each term is computed from the last by dividing by 4.  We do not need to check the other terms because the definition of a geometric sequence is that the quotients are all the same; if that weren't true, it wouldn't be a geometric sequence and the problem would be incorrectly stated.

 

So the next three terms are 25/8, 25/32, and 25/128.

 

 

 

 

Answer 2

Hi, Jan.  A geometric sequence is one where any value in the sequence can be determined using the formula:

a_n=a₁r^(n-1)

 

Where n is the nth term in the sequence, a₁ is the first term (in this case, 800), and r is the rate of change between them.

 

To find r, you simply divide the second term by the first:

200/800 = 0.25.

 

Inserting a₁ = 800 and r=0.25 into the formula above, you have the equation for the sequence:

a_n = 800*0.25^(n-1)

 

You can confirm this is correct by checking against the numbers you already know.  For example, the third term is 50.

a₃ = 800*0.25^(3-1) = 800*0.25² = 50

 

Now, using this formula, you can calculate for each of the next three terms (the fifth, sixth and seventh terms).  OR you can shortcut it by multiplying the previous term by the rate of change, 0.25.

 

(25/2)*0.25 = 25/8

(25/8)*0.25 = 25/32

(25/32)*0.25 = 25/128

Answer 3

Hi Jan;

200/800=1/4

50/200=1/4

(25/2)/50

Let's flip the denominator from 50 into 1/50 and multiply it by the numerator...

(25/2)(1/50)=

25/100=1/4

x₁/(25/2)=1/4

Let's multiply both sides by the denominator of 25/2...

[(x₁)/(25/2)](25/2)=(1/4)(25/2)

[(x₁)/(25/2)](25/2)=(1/4)(25/2)

x₁=25/8

x₂/x₁=1/4

x₂/(25/8)=1/4

Let's multiply both sides by the denominator of 25/8...

[(x₂)(25/8)]/(25/8)=(1/4)(25/8)

[(x₂)(25/8)]/(25/8)=(1/4)(25/8)

x₂=25/32

x₃/x₂=1/4

x₃/(25/32)=1/4

Let's multiply both sides by the denominator of 25/32...

[(x₃)(25/32)]/(25/32)=(1/4)(25/32)

[(x₃)(25/32)]/(25/32)=(1/4)(25/32)

x₃=25/128

 

 

 

 

 

 

Answer 4

r = 1/4

 

  the next 3 terms will be:

 

    25/8 , 25/32, 25/128