The initial step is to try to find a useful factorisation of the general form (n^4 + 1/4).

We will aim to factorise into two factors as below:

(n^2 +mn +1/2)(n^2 –mn +1/2)

Clearly the first terms in each bracket will combine to give the sought after n^4, and the last in each bracket will give the 1/4 we need. The fact that we have +mn and -mn means that the n^3 terms will cancel, as will the n^1 terms. We must then choose a value of m such that the n^2 terms disappear too.

Just showing the n^2 terms, we have (1/2)n^2 +(1/2)n^2 =(m^2)n^2.

m^2 = 1. So m is +/-1 (but since we have +m in one bracket and -m in the other it makes no difference which we choose).

So (n^4 + 1/4) = (n^2 +n +1/2)(n^2 -n +1/2)

We still need some further manipulation before it becomes useful.

If we take the first bracket and rewrite it as:

(n^2 +n +1/4 +1/4)

Then the first three terms form a perfect square:

((n+(1/2))^2 +1/4)

Doing the same for the second bracket we get

((n-(1/2))^2 +1/4)

To see how this is useful, let’s rewrite some of the terms of the initial question in this new form:

(1^4 +1/4) = ((1.5)^2 +1/4)((0.5)^2 +1/4)

(2^4 +1/4) = ((2.5)^2 +1/4)((1.5)^2 +1/4)

(3^4 +1/4) = ((3.5)^2 +1/4)((2.5)^2 +1/4)

The term ((1.5)^2 +1/4) will appear once in the numerator and once in the denominator, and so will cancel out. This will be true for almost all of the terms in the expression and we will just be left with: ((10.5)^2 +1/4)/((0.5)^2 +1/4)

To aid mental calculation, we can rewrite these two terms in the previous form:

(10^2 +10 +1/2)/(0^2 +0 +1/2)

110.5 *2

And so the solution is precisely 221.