Video: How To Solve This Crazy Equation. Ramanujan’s Radical Brain Teaser
Hey, this is Presh Talwalkar. Can you solve for x in the following equation? This problem was posed by Srinivasa Ramanujan in the Journal of the Indian mathematical society over a hundred years ago Give this problem a try and when you're ready keep watching the video for the solution You might try and find an approximation in a calculator a spreadsheet or computer program You'll get that x is equal to about three In fact, x is equal to exactly three Let's see why Three is equal to the square root of nine. Nine equals one plus eight and eight factors out into two times four Four can be expressed as the square root of sixteen We now have a square root inside of another square root.
We've come to the first iteration of this nested radical Can we get to the next stage? Well 16 equals one plus fifteen and fifteen factors out into three times five Five can be re-written as the square root of twenty-five, and we've come to the next iteration of the nested radical 25 equals 1 plus 24 and 24 factors out into four times six Six is the square root of 36, and we've come to the next stage In fact I claim this pattern continues indefinitely Therefore we proved this infinite nested radical is equal to the original term which is three. But how do we know the pattern continues? If you take a look at the final square root, we always have the square root of a perfect square We want to know can we get from this term to the next term? That is if we have the final square root being a perfect square where the coefficient is being multiplied by is 2 less than the perfect square can we always take that perfect square and increment the perfect square and increment the term that is being multiplied by This is true Algebraically we start out with our original term and we expand out the square We can then rewrite this as for equaling one plus three We now factor out n squared plus 4n plus 3 into the terms n plus 1 pimes n plus 3 Now we write a radical where we have the square root of N plus 3 squared.
And this gets us to what we wanted to prove So what any iteration we can always get to the next stage and that shows this pattern will continue Did you figure it out? Thanks for watching this video please subscribe to my channel I make videos on math and game theory you can catch me on my blog Mind Your Decisions which you can follow on Facebook Google+ and Patreon. You can catch me on social media @preshtalwalkar. If you like this video please check out my books. There are links in the video description.