Seeds, constants and squares

Question

Extnd problem:
The purpose of this extended problem is investigate patterns in arithmetic progression and specivically to look for perfect squares.

Begin with whole number from 0 to 10. Call this number the seed. Select another whole number from 1 to 10. Call this number the constant. Generate and arithmetic progression in the following manner:

- the first term is the seed.
- the second term is the sum of the seed and the constant.
-the third term is the sum of the seed and twice the constant.
-and so on, endlessly.....

1. Generate at least 7 different arithmetic progression.
2. Do any of your progression have perfect squares? Predict combinations of seeds and constants that will produce arithmetic progression with no perfect square. Provide mathematical evidence that your pick will never produce a perfect square.

3. Predict combinations of seeds and constants that provide arithmetic progression with exactly one perfect square. Provide mathematical evidence that your pick will produce exactly one perfect square, or that it cannot occurs.
4. What other patterns did you notice while working on this?

Answer 1

 

A progression can generally be expressed as follows.

 

     p_k = s + (k-1)·c, k = 1, 2, 3, ...

 

where s and c are an integer 0 ≤ s ≤ 10, 1 ≤ c ≤ 10.

 

As a perfect square ends with any of 0, 1, 4, 5, 6, or 9,

a number which ends with any of 2, 3, 7, or 8 can not be a perfect square.

 

So, a combination (seed, constant) = (3, 10), for example, will produce a progression with no perfect square.

 

And, a progression with exactly one perfect square can not occur.

 

Hope these help.