Continued Square Roots

 The long ( infinitely … ! ) row of complicated-looking square root signs within this problem put it firmly in the ‘Einstein cartoon ’ category, as far as Maths problems go, but a bit of investigating will reveal pleasant surprises …

 The Problems :

 1.         Find the exact value of the infinite sequence of square roots :


( If you have a calculator with the previous ‘ANS’ button on it, then your working can be repeated automatically by just pressing the ‘=’ button repeatedly once you’ve figured out the sequence of steps you want to do )

 2.         Repeat the problem for : 


You may be feeling happier about these long sequences by now … 

3.         What about   ?


4.         Does             work out nicely ?


5.         Can you find other starting numbers that do yield nice, integer, answers ?


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Last modified: June 18, 2007