How would I word this statement mathematically?

Question

Hi,

If you write out all the solutions for the sum √ [4(n)]  (i.e., the square root of 4 times n), you get a sequence of numbers.
where;  0 <= n <= X,

Most of the numbers in the sequence are irrational but every now and again a whole number pops into the sequence.
An example of the sequence is below. (in the example i have set:
irrational numbers = O
whole numbers = 1)

1,1,O,O,1,O,O,O,O,1,O,O,O,O,O,O,1,...

As you can see from the above example; the whole numbers fall on the:

1st, 2nd, 5th, 10th and 17th (etc) iteration of the formula √ [4(n)].

Therefore the sequence of numbers i end up with from the iteration of √ [4(n)] is:

1,2,5,10,17,...

What I am trying to do is word, mathematically, the process I have just done above? i.e., Im trying to describe, mathematically the following steps:

i) Iterate the formula √ [4(n)]  X amount of times.

ii) From those results, see which of the answers is a whole number.

iii) create and describe a sequence based on the positions of the whole numbers in the sequence of results of √ [4(n)]

************************************************NOTE**********************************************

The sequence of numbers that you get from noting the positions of the whole numbers in the sequence of the results of √ [4(n)], i.e.:

1,2,5,10,17,...

is defined by:

n²+1

but this result isn't exactly what im trying to describe! What I need help with is trying to describe that the whole numbers fall on the 1st, 2nd, 5th etc iteration  of the sum?

Anonymous · Accepted Answer

First, you're not dealing with a sum, you're dealing with a sequence.  I would avoid the word "sum" here since it's neither a sum nor a sequence of partial sums.  Other than that, though, I think your description is just fine, mathematically.

Your conclusion is this:

The (n^2 + 1)-st term of the sequence {sqrt(4n)} is a whole number (non-negative integer) for n = 0, 1, 2, 3, ....

Does that help?

By the way, do you see why this is so?  sqrt(4n) = sqrt(4) sqrt(n) = 2sqrt(n).  This will be rational when n is a perfect square and irrational otherwise.

If we started counting at 1 this would happen at the (n^2)-st terms, but since we're starting at 0 we have it at the (n^2 + 1)-st terms.

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How would I word this statement mathematically?

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