Difference between revisions of "Arithmetic sequence"

Revision as of 22:29, 29 August 2021

In algebra, an arithmetic sequence, sometimes called an arithmetic progression, is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant is called the common difference of the sequence.

For example, $-7, 0, 7, 14$ is an arithmetic sequence with common difference $7$ and $99, 91, 83, 75, \ldots$ is an arithmetic sequence with common difference $-8$ ; However, $1, 2, 3, -4$ and $4, 12, 36, 108$ are not arithmetic sequences, as the difference between consecutive terms varies.

More formally, the sequence $a_1, a_2, \ldots , a_n$ is an arithmetic progression if and only if $a_2 - a_1 = a_3 - a_2 = \cdots = a_n - a_{n-1}$ . This definition appears most frequently in its three-term form; that constants $a$ , $b$ , and $c$ are in arithmetic progression if and only if $b - a = c - b$ .

Properties

Because each term is a common distance from the one before it, every term of an arithmetic sequence can be expressed as the sum of the first term and a multiple of the common difference. Let $a_1$ be the first term, $a_n$ be the $n$ th term, and $d$ be the common difference of any arithmetic sequence; then, $a_n = a_1 + (n-1)d$ .

A common lemma is that given the $n$ th term $x$ and $m$ th term $y$ of an arithmetic sequence, the common difference is equal to $\frac{y-x}{m-n}$ .

Proof: Let the sequence have first term $a_1$ and common difference $d$ . Then using the above result, $\frac{y-x}{m-n} = \frac{(a_1 + (m - 1)d) - (a_1 + (n-1)d)}{m-n} = \frac{dm - dn}{m-n} = d,$ as desired. $\square$

Another lemma is that for any consecutive terms $a_{n-1}$ , $a_n$ , and $a_{n+1}$ of an arithmetic sequence, then $a_n$ is the average of $a_{n-1}$ and $a_{n+1}$ . In symbols, $a_n = \frac{a_{n-1} + a_{n+1}}{2}$ . This is mostly used to perform substitutions.

Sum

Example Problems and Solutions

Introductory Problems

Intermediate Problems

Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$ , given that the roots form an arithmetic progression.

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-== Arithmetic Sequence Video ==
+In [[algebra]], an '''arithmetic sequence''', sometimes called an '''arithmetic progression''', is a [[sequence]] of numbers such that the difference between any two consecutive terms is constant. This constant is called the '''common difference''' of the sequence.
-[https://youtu.be/JMGnPki7PuM Arithmetic Sequence]
-==Definition==
+For example, <math>-7, 0, 7, 14</math> is an arithmetic sequence with common difference <math>7</math> and <math>99, 91, 83, 75, \ldots</math> is an arithmetic sequence with common difference <math>-8</math>; However, <math>1, 2, 3, -4</math> and <math>4, 12, 36, 108</math> are not arithmetic sequences, as the difference between consecutive terms varies.
-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==
+More formally, the sequence <math>a_1, a_2, \ldots , a_n</math> is an arithmetic progression if and only if <math>a_2 - a_1 = a_3 - a_2 = \cdots = a_n - a_{n-1}</math>. This definition appears most frequently in its three-term form; that constants <math>a</math>, <math>b</math>, and <math>c</math> are in arithmetic progression if and only if <math>b - a = c - b</math>.
-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==
+== Properties ==
-{{main|Arithmetic series}}
+Because each term is a common distance from the one before it, every term of an arithmetic sequence can be expressed as the sum of the first term and a multiple of the common difference. Let <math>a_1</math> be the first term, <math>a_n</math> be the <math>n</math>th term, and <math>d</math> be the common difference of any arithmetic sequence; then, <math>a_n = a_1 + (n-1)d</math>.
-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>
+A common lemma is that given the <math>n</math>th term <math>x</math> and <math>m</math>th term <math>y</math> of an arithmetic sequence, the common difference is equal to <math>\frac{y-x}{m-n}</math>.
-or
+''Proof'': Let the sequence have first term <math>a_1</math> and common difference <math>d</math>. Then using the above result, <cmath>\frac{y-x}{m-n} = \frac{(a_1 + (m - 1)d) - (a_1 + (n-1)d)}{m-n} = \frac{dm - dn}{m-n} = d,</cmath> as desired. <math>\square</math>
-<math>\frac{7}{2}(5+17)=77</math>
+Another lemma is that for any consecutive terms <math>a_{n-1}</math>, <math>a_n</math>, and <math>a_{n+1}</math> of an arithmetic sequence, then <math>a_n</math> is the average of <math>a_{n-1}</math> and <math>a_{n+1}</math>. In symbols, <math>a_n = \frac{a_{n-1} + a_{n+1}}{2}</math>. This is mostly used to perform substitutions.
+== Sum ==
 == Example Problems and Solutions ==

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