Difference between revisions of "Arithmetic sequence"
(→Introductory Problems) |
(→Definition) |
||
| Line 1: | Line 1: | ||
| + | == Arithmetic Sequence Video == | ||
| + | https://youtu.be/JMGnPki7PuM | ||
| + | |||
==Definition== | ==Definition== | ||
An '''arithmetic sequence''' is a [[sequence]] of numbers in which each term is given by adding a fixed value to the previous term. For example, -2, 1, 4, 7, 10, ... is an arithmetic sequence because each term is three more than the previous term. In this case, 3 is called the ''common difference'' of the sequence. More formally, an arithmetic sequence <math>a_n</math> is defined [[recursion|recursively]] by a first term <math>a_0</math> and <math>a_n = a_{n-1} + d</math> for <math>n \geq 1</math>, where <math>d</math> is the common difference. Explicitly, it can be defined as <math>a_n=a_0+dn</math>. | An '''arithmetic sequence''' is a [[sequence]] of numbers in which each term is given by adding a fixed value to the previous term. For example, -2, 1, 4, 7, 10, ... is an arithmetic sequence because each term is three more than the previous term. In this case, 3 is called the ''common difference'' of the sequence. More formally, an arithmetic sequence <math>a_n</math> is defined [[recursion|recursively]] by a first term <math>a_0</math> and <math>a_n = a_{n-1} + d</math> for <math>n \geq 1</math>, where <math>d</math> is the common difference. Explicitly, it can be defined as <math>a_n=a_0+dn</math>. | ||
Revision as of 15:03, 23 December 2020
Contents
Arithmetic Sequence Video
Definition
An arithmetic sequence is a sequence of numbers in which each term is given by adding a fixed value to the previous term. For example, -2, 1, 4, 7, 10, ... is an arithmetic sequence because each term is three more than the previous term. In this case, 3 is called the common difference of the sequence. More formally, an arithmetic sequence 
is defined recursively by a first term 
and 
for 
, where 
is the common difference. Explicitly, it can be defined as 
.
Terms in an Arithmetic Sequence
To find the 
term in an arithmetic sequence, you use the formula
![\[a_n = a_1 + d(n-1)\]](http://latex.artofproblemsolving.com/7/f/2/7f2b52ffef55f17744e75c96ec9c9ce85fa69f31.png)
where is the term, 
is the first term, and is the difference between consecutive terms.
Sums of Arithmetic Sequences
- Main article: Arithmetic series
There are many ways of calculating the sum of the terms of a finite arithmetic sequence. Perhaps the simplest is to take the average, or arithmetic mean, of the first and last term and to multiply this by the number of terms. Formally, 
. For example,
or
Example Problems and Solutions
Introductory Problems
- 2005 AMC 10A Problem 17
- 2006 AMC 10A Problem 19
- 2012 AIME I Problems/Problem 2
- 2004 AMC 10B Problems/Problem 10
Intermediate Problems
- Find the roots of the polynomial
, given that the roots form an arithmetic progression.
, given that the roots form an arithmetic progression.
