Sequence of SquaresDate: 07/25/98 at 23:52:11 From: Sam Subject: Geometric sequences I have searched the Internet but cannot find any information specifically discussing or defining the sequence of squares. I would be very grateful for this. Thanks, Sam. Date: 08/04/98 at 16:18:18 From: Doctor Margaret Subject: Re: Geometric sequences Hi Sam, Thanks for writing to us. A sequence of numbers is a set of numbers arranged in an orderly fashion such that the preceding and following numbers are completely specified. For example, in the sequence: ..., 7, 10, 13, 16, 19, ... the difference between the numbers is 3, and this shows us that the number preceding 7 is a 4 and the number after 19 is 22. Take a look at these numbers: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 So far, the righthand column turns out to be the sequence of perfect squares. I think it's neat that this sequence of squares is made up of consecutive odd integers (the lefthand column ). There is also a way to express this algebraically. Odd integers are usually written as (2n - 1) or (2n + 1). The 2 makes sure that the first number is even. The n stands for the number of terms in the sequence, for instance, for three terms starting at n = 1: 3 --- \ 2n-1 = 3^2 = (2(1)-1) + (2(2)-1) + (2(3)-1) = 1 + 3 + 5 = 9 / --- n = 1 You don't have to start at 1. You can start anywhere and go on for however many terms you like or are required to. Let me know if you need anything else on this subject. Thanks for writing. - Doctor Margaret, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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