q = sqrt(N+k) - sqrt(N-k); An Interesting Mathematical problem: Maths is Fun

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After publishing my blog yesterday on an interesting mathematical problem, I realized that a simple variation of the problem is equally interesting and worth publishing too.  This I do in the following.

(Click on the slide to see its full page image)

Equations 3, 4 and 5 provide the full solution of the problem.  Only even values of q are possible and their range is defined by eqns. 3 and 4.  Eqn.5 may then be used to calculate the corresponding values of k. 

The next slide shows the allowed values of q.

The range of q is seen to be quite small - in fact for N < 50, the number of allowed q values is 4 but increases as N gets bigger .  

For example, for N = 30, q(min) = 0 and q(max) = 7.75.  Since q is an even integer, only possible values of q are 2, 4 and 6.  The corresponding values of k are (use eqn.5) 10.77, 20.40 and 27.50. 

Final Word:  I find the problem to be very instructive - particularly, it demonstrates that some serious conclusions may be obtained by initial observations of the problem.  For example, by looking at the expression in eqn.1, we could establish the range of q values - later confirmed by the general analysis.  

Plotting a graph of the results is also very useful as this provides a wholesome view of the results obtained.

I hope that you enjoyed our little excursion and agree that 'Maths is Fun'.