IB MATH SL
Introduction: In this task I will consider a set of numbers that are presented in a symmetrical pattern and try to find a general equation to find the elements in the [pic]row. Consider the five rows of number shown below.
Figure 1 Lacsap’s Fractions
The aim of this task is to find the numerator of the sixth row and to find the general statement for [pic]. Let [pic] be the [pic]element in the [pic]row, starting with r=0.
First, I will make a table of the numerator and the row number to show the relationship between the numerator and the row number.
Table 1 A table showing the relationship of rows and numerator The difference between the numerator of row 1 and row 2 is 2, then the difference between the numerator of row 2 and row 3 is 3. The second difference for each row is 1 and it shows that this is a geometric sequence. So, I will start by finding the equation using the quadratic formula, [pic], in which x is the row number and y is the numerator. First, I will plug in the numbers in the second row, which is 2 for x and 3 for y to try finding the quadratic equation of the Lacsap’s sequence. Then, I will plug in the numbers in the third row and it forms a simultaneous equation. I will use substitution method to solve the equation as there are two unknowns in the equation. First, I will solve for ‘a’ by substituting the answer ‘b’ from the first equation, which is row 2, to the second equation, which is row 3. The workings are shown below. The answer of ‘a’ and ‘b’ are both 0.5. So, we found out the equation for the numerator is [pic], where n is row number. [pic][pic]
The aim of this task is to find out the numerator of the sixth row. So, I plugged in row number in the equation to find the numerator of the sixth row. [pic]
From this method, we found out the numerator of the sixth row should be 21. In order to test the...