Date: 04/16/2001 at 10:25:46 From: Steve Subject: Millionth Term What is the millionth term in the following sequence? 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...
Date: 04/16/2001 at 16:23:52 From: Doctor Rob Subject: Re: Millionth Term Thanks for writing to Ask Dr. Math, Steve. This sequence S(n) is connected to the triangular numbers T(k): 1, 3, 6, 10, ... by the fact that S(T(k)) = k. In fact, what you need to do to find S(1,000,000) is to find a value k such that T(k-1) < 1,000,000 <= T(k), and then k will be your answer. Now you know that T(k) = k*(k+1)/2, so you have two simultaneous quadratic inequalities to solve: k*(k+1) - 2000000 >= 0 k*(k-1) - 2000000 < 0 I would use the quadratic formula to solve each. The first will give you a lower bound on k; the second will give you an upper bound on k. The true value of k will be the integer lying between these bounds. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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