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Nice summation puzzle

Chris Smith (@aap03102) runs a weekly maths newsletter during termtime. If you’re not on the list, I’d advise you to get on it (drop him a line in twitter). It has some great stuff in it, such as this fraction:


How ace is that?

He also has a puzzle of the week feature. This week’s looked cool, so I had a crack at it:


I thought about it a little, and saw that it boiled down to this:


I had the general term: 1/((n^1/2)+(n+1)^1/2) but I wasn’t sure where that would get me. I tried rationalising the denominator of the first two terms:


Now I could see where I was getting to, I thought I’d check the general term, to be sure:


Then it was a case of looking at the sum:


The sum of (n+1)^1/2 -n^1/2 for n=1 to n=91808

Then I cancelled:


Leaving just

91809^1/2 -1^1/2

Which is

303-1 =302

A lovely neat solution to a lovely summation problem which I hope to try in my sixth formers soon, and which I hope will work well with the new GCSE specifications.

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