1.Introduction 3
2.Deriving general expression of Numerator 3
I.Using Pattern Recognition 3
II.Using Progression 5
3.Deriving general expression of denominator 10
I.Using Pattern Recognition 10
II.Using Progression 11
4.Validating the general expression 14
5.FINDING ADDITIONAL ROWS USING GENERAL EXPRESSION 19
6.Conclusion 36
7.Limitations 36
1. Introduction
Lacsap's fractions are based upon Pascal's triangle. The general triangle's rule is "Starting with1/1 at the top, the numerator (top number) is increased by one each step to downward right, and the denominator (bottom number) is increases by one each step to the downward left. Each fraction on the inside thereby has the numerator of the fraction to its upper-right, and the denominator of the fraction to its upper-left." So basically, going to the right, the denominator increases but if we go to the left, the numerator increases, both by one.
The purpose of this assignment is to examine a set of fractions presented in a symmetrical pyramid, and generate a general formula for the fractions with respect to the row and element number after considering the first five rows. We'll be finding a general statement for En(r) of the Lacsap's fractions, where En(r) denotes the fraction, n stands for the nth row and r stands for the number in the row starting with r = 0. I am going to use MS-Office as a technological tool.
Let's represent each of the Lacsap's fraction in the following form:
where, n denotes the row number and r denotes the element number in the row starting from 0.
Let N(n) represents the numerator of the fraction and denotes the denominator. Before trying to find a general statement, we should change all the 1's into fractions based on the numerator because all the numerators of each row are the same. Our pattern should look like this after doing so.
Now we can separate the numerator from the denominator and work with it separately to get t...