Question 536481
3rd and 5th term of a G.P are 25 and 100 respectively.  Find the 7th term:
<pre>
    ___, ___,  25, ___, 100, ___, <u> ? </u>

We could use formulas but the easiest way is 

1. Let r = the common ratio,
2. the 4th term has to equal the 3rd term, 25, times r. So fill in the
   3rd term as 25r:

    ___, ___,  25, 25r, 100, ___, <u> ? </u>
  
3. Then the 5th term has to equal the 4th term, 25r, times r. 

4. Get the equation from 

                          {{{(matrix(2,1,4th, term))}}}×r = {{{(matrix(2,1,5th, term))}}}
 
                           (25r)r = 100
                             25r² = 100
                               r² = 4
                                r = ±2

Aha!  There are two solutions for r, so we will use ±r for both:

5. Get the 6th term by multiplying the 5th term, 100 by ±2 getting ±200  

    ___, ___,  25, 25r, 100, ±200, <u> ? </u>

6. Get the 7th term by multiplying the 6th term, ±200, by ±2, and
regardless of whether r=+2 or r=-2, the 7th term will be +400

    ___, ___,  25, 25r, 100, ±200, 400
 
If we like we can get the whole progressions:

One way the progression could be is this:

    6.25,  12.5, 25,  50, 100,  200, 400

and the other way is

    6.25, -12.5, 25, -50, 100, -200, 400
    
But either way the 7th term is 400.

Edwin</pre>