Geometric Sequence Problem Help!

Question

Let x-2,x,x+3 be the first 3 terms of a geometric sequence. Find the value of x.

Answer 1

Since you have a geometric sequence, each consecutive term is derived by multiplying by a ratio. For instance, if my geometric sequence had 1/2, 1/4, and 1/8 as the first three terms, my ratio would be 1/2. (1/2 * 1/2 = 1/4,  1/4 * 1/2 = 1/8). The find the ratio, you just divide a term by it's previous term. Using my previous example, 1/4 divided by 1/2 = 1/2. For your problem, you will have x/(x-2) = k and (x+3)/x = k. k is just an arbitrary letter we can use since we don't actually know the ratio that was used for this geometric series. However, we know that each ratio has to be the same, so both equations must equal k. And since they both equal k, you can set the two equations equal to each other:

  x    =   x+3 

 x-2         x

 

Now it's just using algebra to solve for x:

x^2 = (x+3)(x-2)

x^2 = x^2 + x - 6

0=x - 6

x=6

 

Check the answer:

x - 2 = 6 - 2 = 4

x = 6

x + 3 = 6 + 3 = 9

 

 6  =  3 

 4      2

 

 9  =  3 

 6      2

Ratios match!