Difference between revisions of "Arithmetic sequence"
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==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>. | ||
+ | |||
+ | ==Terms in an Arithmetic Sequence== | ||
+ | To find the <math>n^{th} </math> term in an arithmetic sequence, you use the formula | ||
+ | <cmath>a_n = a_1 + d(n-1)</cmath> | ||
+ | where <math>a_n</math> is the <math>n^{th}</math> term, <math>a_1</math> is the first term, and <math>d</math> is the difference between consecutive terms. | ||
==Sums of Arithmetic Sequences== | ==Sums of Arithmetic Sequences== | ||
+ | {{main|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, <math>s_n=\frac{n}{2}(a_1+a_n)</math>. For example, | ||
− | + | <math>5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math> | |
− | + | or | |
+ | <math>\frac{7}{2}(5+17)=77</math> | ||
== Example Problems and Solutions == | == Example Problems and Solutions == | ||
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* [[2005_AMC_10A_Problems/Problem_17 | 2005 AMC 10A Problem 17]] | * [[2005_AMC_10A_Problems/Problem_17 | 2005 AMC 10A Problem 17]] | ||
* [[2006_AMC_10A_Problems/Problem_19 | 2006 AMC 10A Problem 19]] | * [[2006_AMC_10A_Problems/Problem_19 | 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 <math>x^5-5x^4-35x^3+ax^2+bx+c</math>, given that the roots form an arithmetic progression. | ||
==See Also== | ==See Also== |
Revision as of 18:27, 19 January 2018
Contents
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 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.