A Hundred-Row Number Pyramid
Date: 11/19/98 at 11:55:42 From: Terry Williams Subject: Row, row, row your boat Hello, I am a secondary math teacher and I am having difficulty with the following question. Given the following 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 What is the last number (on the righthand side) in the one-hundredth row? Could you please send me an outline solution? I would appreciate it. Thanks in advance, Terry
Date: 12/01/98 at 19:54:37 From: Doctor Ameis Subject: Re: Row, row, row your boat Hello Terry, Here is one approach to your problem. The approach involves setting up an input/output table and analyzing it to see what kind of relation exists between the row number (the input) and the rightmost number in a row (the output). Here is the table: input(row number) | output(rightmost number) ------------------+------------------------- 1 | 2 2 | 5 3 | 9 4 | 14 5 | 20 100 | ? If you figure out the difference between consecutive output values, these are +3 (5 - 2), +4 (9 - 5), +5 (14 - 9), +6 (20 - 14). These differences increase each time by a constant amount of +1. This means that the relation between the input and output values is a quadratic function (output = a * (input)^2 + b * (input) + c OR using more conventional notation: y = ax^2 + bx + c). Pick four pairs of data (three pairs would do), construct the quadratic function for each pair, and solve the resulting equations for a, b, and c. This will give you the relation between the input and output that you need to determine the rightmost number in the hundredth row. Here is one possible way to proceed: 14 = a*(4)^2 + b*(4) + c 2 = a*(1)^2 + b*(1) + c Simplifying and subtracting the two equations we get: 12 = 15a + 3b 20 = a*(5)^2 + b*(5) + c 9 = a*(3)^2 + b*(3) + c Simplifying and subtracting the two equations we get: 11 = 16a + 2b Solving these two equations for a and b, and then using that information to solve for c, we get: a = 1/2, b = 3/2, c = 0 This means that one way to express the relation between input (the row number) and output (the rightmost number in row) is: output = 1/2 * (input)^2 + 3/2 * (input) Another way to arrive at the relation is examine the input/output table and use "educated" trial and error to figure out the relation between the row number and the rightmost number. Doing it this way, you might see that: rightmost number = 1/2 * (row number)(row number + 3). Hope this helps. Feel free to ask for more assistance if needed. - Doctor Ameis, The Math Forum http://mathforum.org/dr.math/
Date: 12/01/98 at 19:59:13 From: Doctor Stacey Subject: Re: Row, row, row your boat Hi Terry, If you're explaining the solution to a class, I think Dr. Ameis's second method may be easier. It comes from realizing that the last number in each row (and in fact any number) is the number of numbers up to and including it. So you just need to count the number of numbers in 100 rows. There are two in the first row, three in the second, ..., 101 in the hundredth. Remember how to add 1 through 100 really quickly? Use the same method. First plus last, times the number of rows, divided by 2. Hope that helps! - Doctor Stacey, The Math Forum http://mathforum.org/dr.math/
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