Challenge of the Week: Interesting Recursive Formula - Data Science Central 2020-08-13T12:24:56Z https://www.datasciencecentral.com/forum/topics/challenge-of-the-week-interesting-recursive-formula?feed=yes&xn_auth=no You can prove this with some… tag:www.datasciencecentral.com,2016-02-19:6448529:Comment:389275 2016-02-19T18:01:49.046Z Bryan M. Gorman https://www.datasciencecentral.com/profile/BryanMGorman <p>You can prove this with some straightforward algebra. Define a(n) = S(n) - S(n-1) for n&gt;=2. The sum follows immediately, and it's easy to show that a(n+1) = R(n)a(n) by the recursive definition of S(n). This product telescopes back to n=2, though, not n=1. R(n) is not defined for n=1; otherwise, you would have to define S(0).</p> <p>You can prove this with some straightforward algebra. Define a(n) = S(n) - S(n-1) for n&gt;=2. The sum follows immediately, and it's easy to show that a(n+1) = R(n)a(n) by the recursive definition of S(n). This product telescopes back to n=2, though, not n=1. R(n) is not defined for n=1; otherwise, you would have to define S(0).</p>