Question

If the sequence −1 1/3 , 4, k, 36 is geometric, find the value of k.

Answer 1

Answer: k = -12

The mixed number -1 & 1/3 converts to the improper fraction -4/3

Let r be the common ratio. To go from one term to the next, we multiply by this common ratio. So,

second term = (first term)*(common ratio)

4 = (-4/3)*r

3*4 = 3*(-4/3)*r

12 = -4r

-4r = 12

r = -3

We multiply each term by -3 to get the next term. The third term is therefore,

third term = (second term)*(common ratio)

third term = 4*r

third term = 4*(-3)

third term = -12

and if we keep going...

fourth term = (third term)*(common ratio)

fourth term = -12*(-3)

fourth term = 36

So it matches up

Answer 2

If a, b, c are the geometric sequence, then

`ac=b^2`

We have a = 4, b = k, c = 36. Substitute:

`(4)(36)=k^2\\\\k^2=144\to k=\pm√(144)\to k=-12\ \vee\ k=12`

First term is negative, therefore it's the alternating sequence. Therefore your answer is

k = -12