SOLUTION: What are the next five terms in the sequence: 2, 1/2, 2/9, 2/16, 2/25.....?

What are the next five terms in the sequence:
2, 1/2, 2/9, 2/16, 2/25.....?

There are two methods to try.  Successive
differences and successive ratios (quotients).
One method is usually easier than the other.

First I'll try doing it by successive differences 
to see if there is an easily recognizable 
pattern:

2nd term - 1st term = 1/2 - 2 = 1/2 - 4/2 = -3/2

3rd term - 2nd term = 2/9 - 1/2 = 4/18 - 9/18 = -5/18

4th term - 3rd term = 2/16 - 2/9 = 1/8 - 2/9 = -7/72 

5th term - 4th term = 2/25 - 2/16 = 2/25 - 1/8 = 
                            16/200 - 25/200 = -9/200

The numerators are successive negative odd numbers,
but the denominators do not form a very easily 
recognizable pattern. Maybe we could find a recognizable 
pattern for them, but it's not immediately obvious.  
So let's turn to the other method, the successive ratios
(quotients) to see if it is any easier.  If not we will 
have to come back to this one and try to find a 
recognizable pattern for those denominators.

So let's check the successive ratios
(quotients) to see if there is a more 
easily recognizable pattern:

2nd term ÷ 1st term = 1/2 ÷ 2 =  1/2 × 1/2 = 1/4

3rd term ÷ 2nd term = 2/9 ÷ 1/2 =  2/9 × 2/1 = 4/9

4th term ÷ 3rd term = 2/16 ÷ 2/9 =  2/16 × 9/2 = 9/16

5th term ÷ 4th term = 2/25 ÷ 2/16 =  2/25 × 16/2 = 16/25

Yes we can easily recognize that pattern, because the
numerators and denominators are the successive perfect
squares.

1/4 = 1²/2²

4/9 = 2²/3²

9/16 = 3²/4²

16/25 = 4²/5²

so we can extend the successive ratios

1²/2², 2²/3², 3²/4², 4²/5²

this way:

1²/2², 2²/3², 3²/4², 4²/5², 5²/6², 6²/7², 7²/8², 8²/9², 9²/10²

These are

1/4, 4/9, 9/16, 16/25, 25/36, 36/49, 49/64, 64/81, 81/100

So the sequence

2, 1/2, 2/9, 2/16, 2/25

can be extended to the next five terms this way:

6th term = 5th term × 25/36 = 2/25 × 25/36 = 1/18

7th term = 6th term × 36/49 = 1/18 × 36/49 = 2/49

8th term = 7th term × 49/64 = 2/49 × 49/64 = 1/32

9th term = 8th term × 64/81 = 1/32 × 64/81 = 2/81

10th term = 9th term × 81/100 = 2/81 × 81/100 = 1/50

So the answer is:

2, 1/2, 2/9, 2/16, 2/25, 1/18, 2/49, 1/32, 2/81, 1/50

In doing such problems, we should try both methods,
to see which has the more easily recognizable pattern.

Edwin