Sequence Investigation Essay, Research Paper

There are three types of Bugs in this investigation : Happy BugSad BugBlank Bug

Each bug splits into two to make new bugs. These are the splitting rules : Sad Bug

Happy Bug

Blank Bug

Happy Bug

Sad Bug

Happy Bug

Happy Bug

Blank Bug

Sad Bug All the bugs live the same length of time. Bugs die when they split. The different generations can be shown as follows: In order to be able to expand the generations into the 4th and 5th , I will simplify the bugs into letters. An H symbolises a Happy Bug An S symbolises a Sad Bug A B symbolises a Blank Bug. I have discovered that from the 4th and 5th generations onward, the pattern continues as shown below. These generations are shown in full in the appendix. My results can be shown in a table such as the one below, which gives the numbers of different bug types in successive generations:

GENERATIONHAPPYSADBLANKTOTAL

11001

20112

33104

42338

595216 From the table, I can see that the number of Happy Bugs in a given generation is twice the number of Sads added to the number of Blanks, both from the previous generation. This is because each Sad in the previous generation turns into two Happys (so it the Sads times two) and each Blank in the previous generation splits to produce one Happy (so this is the same number as the Blanks). The same can be performed on the Sad Bugs i.e. the number of Sads in a given generation is equal to the number of Happys plus Blanks in the previous generation. Doing the same for Blank Bugs shows that the following equations can be applied: H2 = 2 ( S1 ) + B1

S2 = H1 + B1 = B2 + B1 (from below)

B2 = H1

Where H2 = Happys in a generation

H1 = Happys in the generation before and so on. From the table above, I can work out the probabilities of picking a given type of bug from a specified generation. This is the same as the proportion of that bug occurring within the generation.