Question

How do you find these missing terms [__,8,__,128,....]?

Answer 1

There are two solutions:

`4` and `32`

Or:

`-4` and `-32`

Explanation:

If the general term of the sequence is `a_n` (`n = 1, 2, 3,...`) then we can rephrase the problem like this:

A geometric sequence has `a_2 = 8` and `a_4 = 128`. What are `a_1` and `a_3` ?

The general term of a geometric sequence is described by the formula:

`a_n = a*r^(n-1)`

where `a` is the initial term and `r` is the common ratio.

In our example, we find:

`r^2 = (a r^3)/(a r) = a_4/a_2 = 128/8 = 16 = 4^2`

So `r = +-4`.

If `r = 4` then:

`a_1 = a_2/r = 8/4 = 2` and `a_3 = a_2*r = 8*4 = 32`

If `r = -4` then:

`a_1 = a_2/r = 8/(-4) = -2` and `a_3 = a_2*r = 8*(-4) = -32`