A sequence is a collection of numbers in a defined order. The terms of a sequence follow a certain rule.

E.g. 2, 5, 8, 11, .... rule: start at 2 and add 3 to each consecutive number. This is an arithmetic sequence.

0, 3, 6, 12, 24, .... rule: start at 3 and double each term to produce the next term. This is a geometric sequence.

The sum of all the terms in a sequence is a series. A series can have a finite or infinite number of terms.

$1 + 1/2 + 1/3 + 1/4 + 1/5$ .... is the harmonic series.

The sum of a series is symbolised by the capital Greek letter sigma (Σ). Limits may be set for a finite sum:

$$∑↙{i=1}↖n 1/i$$This is shorthand for: $1/1 + 1/2 + 1/3 + ... + 1/n$

Leonard Euler was the first to use this notation for series.

A sequence or series which has a constant common difference, d, between two consecutive terms is arithmetic, with the general term:

$$u_n = u_{1} + (n - 1)d$$where $u_1$ is the first term and d is the common difference.

The sum of a finite arithmetic series is:

$$S_n = n/2[2u_1 + (n - 1)d] = n/2(u_1 + u_n)$$where *n* is the number of terms in the series, $u_1$ amd $u_d$ are the first and last terms, and each term is separated by a common difference, *d*.

If the ratio of two consecutive terms in a sequence or series is constant, then it is geometric.

For the sequence 1, 3, 9, 27, ... ; $u_1 = 1$ is the first term, and r = 3 is the common ratio. The recursive equation is: $u_n = u_{n-1}⋅r$.

The general term is: $u_n = u_1⋅r^{n-1}$, r ≠ -1, 0, 1.

The sum of a geometric series is:

$$S_n = {u_1(1 - r^n)}/{1 - r}, r ≠ 1$$where n is the number of terms in the series, $u_1$ the first term, and r is the common ratio of any two consecutive terms.

Content © Andrew Bone. All rights reserved. Created : October 12, 2014

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

and many more articles in the subject:

Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info.

1929 - 2015

Maurice Strong was a man of extraordinary achievement and a dedicated environmentalist, despite his early life's achievement of rags-to-riches, making his fortune from the oil industry.

Website © contentwizard.ch | Designed by Andrew Bone