Difference between revisions of "Arithmetic sequence"
(→Introductory Problems) |
|||
| (15 intermediate revisions by 8 users not shown) | |||
| Line 1: | Line 1: | ||
==Definition== | ==Definition== | ||
| − | An arithmetic sequence is a sequence of numbers | + | 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 == | |
| + | === Introductory Problems === | ||
| + | * [[2005_AMC_10A_Problems/Problem_17 | 2005 AMC 10A Problem 17]] | ||
| + | * [[2006_AMC_10A_Problems/Problem_19 | 2006 AMC 10A Problem 19]] | ||
| + | * [[2012 AIME I Problems/Problem 2]] | ||
| + | * [[2004 AMC 10B Problems/Problem 10]] | ||
| − | <math> | + | === 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== | ||
| − | [[geometric sequence|Geometric Sequences]] | + | *[[sequence|Sequence]] |
| + | *[[series|Series]] | ||
| + | *[[geometric sequence|Geometric Sequences]] | ||
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
![\[a_n = a_1 + d(n-1)\]](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.
