In the infinite series: 1 + 3 + 3 + 3 + 5 + 5 + 5 + 5 + 5 + 7 + ... each positive odd integer k appears k times in consecutive order.

1) Write a general formula that can be used to find the n_th term of the series.

2) Prove that if n is not 3, then the n_th partial sum of the series is never prime.

(In reply to Part 1 (spoiler) by Leming)

I came up with the exact same formula. To turn the series into 1, 2, 2, 2, 3, 3, 3, 3, 3, 4 ... all one has to do is take the square root and round up, since a new odd number starts every perfect square term. Then simply multiply them by 2 and subtract 1

 

Posted by Gamer on 2007-01-08 13:46:27