Find the 276th LetterDate: 08/15/2001 at 15:06:49 From: Cale McMellon Subject: Sequence pattern What is the pattern for this problem? I know the answer is g, but isn't there an easier way to find it than writing it out? Find the letter that is the 276th entry in the following sequence: g,l,g,l,l,g,l,l,l,g,l,l,l,l,g,l,l,... Date: 08/15/2001 at 15:37:53 From: Doctor Greenie Subject: Re: Sequence pattern Hello, Cale - Here are the numbers of the terms in the sequence that are g: 1, 3, 6, 10, 15, ... In other words, entries 1, 3, 6, 10, 15, ... are g, while the rest are l. These you may recognize as the sums of the first n integers... 1 = 1 3 = 1+2 6 = 1+2+3 10 = 1+2+3+4 15 = 1+2+3+4+5 and so on. You may also be familiar with the formula for the sum of the first n integers: n(n+1) sum = ------ 2 Thus, if this expression has the value 276 for some integer n, then 276 is the sum of the first n integers, and so the 276-th term in your sequence will be g. Let's see if this expression has the value 276 for some integer n: n(n+1) ------ = 276 2 n^2+n = 552 n^2+n-552 = 0 (n+24)(n-23)=0 n = -24 OR n = 23 n = -24 doesn't make sense in this problem, since n is the number of terms in a sequence. The correct answer is n = 23. So the sum of the first 23 integers is 276: 1+2+3+...+22+23 = 276 and, therefore, the 276-th term of your sequence is g. Please write back if you have any further questions on this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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