Mathematics Portfolio Sl

Topics: Triangular number, Geometric progression, Sequence Pages: 18 (2806 words) Published: August 23, 2013
Aga Khan Mzizima Secondary School
Portfolio Type 1: INVESTIGATION Title: SEQUENCES Mathematics Standard Level Teacher: Mr.Lazaro Name: Fatema Ismailjee IB 1 - 2011

Sequence is a set of things (usually numbers) that are in order.

e.g. 1, 2, 3, 4,...
Where 1 is the first term, 2 is the second term and so on. (...)in the end means that the sequence goes on forever. Three dots in the middle e.g. 1, 2, 3...7, 8 indicate that the pattern continues until the next number appears.

There is finite and infinite sequence, infinite sequence is when the sequence has no end and finite is a set with a function e.g. {1, 3,...n} Calculating specific terms leads to an "nth term formula."

Before creating a rule of calculation, you need to realize that sequences are functions with the specific domain of the counting numbers {1, 2, 3, 4, 5, ...}. So the n replaces x as the input variable and instead of writing y, we use an as the output variable. Arithmetic sequence: the difference between one term and the next is a constant in arithmetic sequence. The general formula is an = a1 + (n - 1) d Geometric sequence: A geometric sequence is a group of numbers where each term after the first is found by multiplying the previous one by a fixed non zero number called common ratio. The general formula is an = a1 × rn-1 Series is the sum of terms of a sequence. Sn = x1 + x2 +.......xn Arithmetic series: The general formula is Sn = n/2(a1 + an) Geometric series: a series which has a constant ratio between terms. The general formula is Sn = a1 (1 – rn) 1 - r

TRIANGULAR NUMBERS

Triangular number is the number of dots in an equilateral triangle uniformly filled with dots. This is an investigation task whereby I will try to find number of shapes of geometric figures which form triangular numbers. I will use different sources of information to attain shapes and figures. For the calculations required, different math techniques will be used for the different shape obtained.

Aim

In this task I will consider geometric shapes which lead to special numbers. The simplest examples of these are square numbers, 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3 and 4. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1, 3, 6, ...).

.. . . .
. . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . .
1 3 6 10 15

There is a sequence of the number of dots in the triangular shape above.

Complete the triangular sequence with three more terms.

.
. .
. . .
. . . .
. . . . .
. . . . . .
21 dots

.
. .
. . .
. . . .
. . . . .
. . . . . .
. . . . . . .
28 dots

.
. .
. . .
. . . .
. . . . .
. . . . . .
. . . . . . .
. . . . . . . ....

References: "Sequences." Math Is Fun - Maths Resources. Web. 12 Mar. 2011. 

"Help for a Generic Formula for a Stellar Pattern.? - Yahoo! Answers." Yahoo! Answers - Home. Web. 12 Mar. 2011. 

"Triangular Number - ENotes.com Reference." ENotes - Literature Study Guides, Lesson Plans, and More. Web. 11 Mar. 2011.
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