Lascap's Fractions. I was able to derive a general statement for both the numerators and denominators, and prove it for other rows of the pattern

Authors Avatar by elisabettasorrentino13 (student)

Elisabetta Sorrentino

Lacsap’s Fractions

Math SL Internal  Assessment TYPE I

 

Name: Elisabetta Sorrentino

Subject: Math

Candidate Number: 001459 - 048

School: Suzhou Singapore International School

Pages: 14

Lacsap’s Fractions

Aim: In this task I will consider a set of numbers that are presented in a symmetrical pattern. I will attempt to find the next 2 rows of the pattern and a general statement to find any term. Further more, I will also examine the validity, scope and limitations of my general statement.

As shown on the diagram, the numerator of the fraction increases as we go down the rows. The pattern is straightforward; it is an addition of the row number to the numerator of the previous row.  The amount added is increased by one each row. This pattern can be easily understood in the table below.

Both the numerator and denominator have a consistent pattern when the 1’s from the first row are not considered. Since the first row is not considered, the pattern consequently starts from row 2.

Using Microsoft Excel, I plotted the relation between the row number, n, and the numerator in each row.

From the graph above, I noticed that the trend line of the relation between the row number and the numerator in each row is quadratic. To represent this graph in a general statement, I used a graphic calculator to derive the general statement using the quadratic regression function, following these steps.

  1. Click Stat, then select “1:Edit”

  1. Put the row number in L1 and the numerator of each row on L2
Join now!

  1. Then click STAT, select CALC, then select “5:QuadReg”

  1. Click 2ND, input L1, then comma, input L2

  1. Then comma again, click VARS, then select Y-VARS, select “1:Function…”

  1. Then select “1:Y1

  1. The answer is:

Let x be the row number and y the numerator

SIMPLIFYING

SIMPLIFYING

Therefore the general statement representing the numerator in each row is:

The denominators in each row were ...

This is a preview of the whole essay