SEQUENCE
* In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers. * For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context. ARITHMETIC SEQUENCE
* A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n.
Explicit Formula: an = a1 + (n – 1)d
Example 1: 3, 7, 11, 15, 19 has a1 = 3, d = 4,
and n = 5. The explicit formula is
an = 3 + (n – 1)·4 = 4n – 1
Example 2: 3, –2, –7, –12 has a1 = 3, d = –5, and n = 4. The explicit formula is
an = 3 + (n – 1)(–5) = 8 – 5n
GEOMETRIC SEQUENCE
* Before talking about geometric sequence, in math, a sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.
For examples, the following are sequences:
2, 4, 8, 16, 32, 64, .......
243, 81, 27, 9, 3, 1, ...............
A geometric sequence is a sequence where each term is found by multiplying or dividing the same...
...to study a new subject which
he called calculus. He said that this subject is a foundation of higher
mathematics and that it is going to be very difficult. We have already
studied real numbers, the real line, infinite numerical sequences, and
limits of sequences. The professor was indeed right saying that comprehension of the subject would present difficulties. I listen very
carefully to his explanations and during the same day study the
relevant pages...
...Week One Assignment
Allana Robinson
MAT 126
Survey of Mathematical Methods
Melinda Hollingshed
August 21, 2011
Arithmetic Sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount. This amount is known as the common difference and can be found using a specific formula by substituting the numbers from the word problem into the equation. When you plug in all the information, you are able to...
...Sequences and Convergence
Let x1 , x2 , ..., xn , ... denote an infinite sequence of elements of a metric space
(S, d). We use {xn }∞
n=1 (or simply {xn }) to denote such a sequence.
Definition 1 Consider x0 ∈ S. We say that the sequence {xn } converges to x0
when n tends to infinity iff: For all > 0, there exists N ∈ N such that for all
n > N , d(xn , x0 ) <
We denote this convergence by lim xn = x0 or simply xn −→ x0 .
n→∞
Example 2...
...Anatolia College 
Mathematics HL investigation

The Fibonacci sequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence...
...Arithmetic Sequence shows "Survey of Mathematical Methods" and contains solutions on the following problems:
First Problem: question 35 page 230
Second Problem: question 37 page 230
Mathematics  General Mathematics
Week One Written Assignment
Following completion of your readings, complete exercises 35 and 37 in the “Real World Applications” section on page 280 of Mathematics in Our World .
For each exercise, specify whether it involves an...
...the graph that the pattern/structure is exponential. This is due to the previous numbers being added in succession with the next, resulting in the ‘gap’ between each number to increase.
The trend in which the numbers follow is called a Fibonacci sequence and is often found in nature as well.
Many instances in which the Fibonacci Series is present in nature are that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers....
...Questionbank
Topic 1. Sequences and Series, Exponentials and The Binomial Theorem
1. Find the sum of the arithmetic series
17 + 27 + 37 +...+ 417.
2. Find the coefficient of x5 in the expansion of (3x – 2)8.
3. An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series.
4. Find the coefficient of a3b4 in the expansion of (5a + b)7.
5. Solve the equation 43x–1 = 1.5625 × 10–2.
6. In an arithmetic...
...particularly that of the Fibonacci sequence and the Golden Ratio. In Debussy’s Nocturne, composed in 1892, I look into the use of the Fibonacci sequence and the Golden Ratio. Previously it has been noted that composers used the Fibonacci sequence and the Golden Ratio in terms of form, however in my analysis I look into the use of it in terms of notation as well. I will explore how the idea of Sonata form is used along with the Mathematical Model of...