Arithmetic Progression (A.P.)
A sequence $\displaystyle a_{1},a_{2},a_{3},a_{4},...a_{n},...$ is called arithmetic sequence or arithmetic progression if $\displaystyle a_{n+1}=a_{n}+d,n\epsilon N$ where $\displaystyle a_{1}$ is called the first term and the constant $\displaystyle d$ is called the common difference of A.P.
Terms of an A.P.
$\displaystyle a$=first term
$\displaystyle a_{n}=n^{th}$ term or general term
$\displaystyle l$=last term
$\displaystyle d$=common difference
$\displaystyle n$=number of terms
$\displaystyle S_{n}$=sum of n terms of an A.P.
Let $\displaystyle a$ be the first term and $\displaystyle d$ be the common difference of an A.P., then the A.P. is $\displaystyle a,a+d,a+2d,a+3d...$
$\displaystyle a_{n}=\displaystyle a+(n-1)d$
$\displaystyle l=a+(n-1)d$
$\displaystyle S_{n}=\frac{n}{2}[2a+(n-1)d]$ OR
$\displaystyle S_{n}=\frac{n}{2}[a+l]$
Note:
i) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
ii) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
iii) If each term of an A.P. is multiplied by a constant, the resulting sequence is also an A.P.
iv) If each term of an A.P. is divided by a non-zero constant, the resulting sequence is also an A.P.
v) If the sum of n terms of an A.P. is denoted by $\displaystyle S_{n}$, then its common difference =$\displaystyle S_{2}-2S_{1}$.
vi) The sum of the terms equidistant from the beginning and the end of an A.P. is always the same and equals to the sum of the first and the last terms.
Note: If the sum of the numbers is given, then in an A.P.,
a) three numbers are taken as $\displaystyle a-d,a,a+d$
b) four numbers are taken as $\displaystyle a-3d,a-d,a+d,a+3d$
c) five numbers are taken as $\displaystyle a-2d,a-d,a,a+d,a+2d$
A sequence $\displaystyle a_{1},a_{2},a_{3},a_{4},...a_{n},...$ is called arithmetic sequence or arithmetic progression if $\displaystyle a_{n+1}=a_{n}+d,n\epsilon N$ where $\displaystyle a_{1}$ is called the first term and the constant $\displaystyle d$ is called the common difference of A.P.
Terms of an A.P.
$\displaystyle a$=first term
$\displaystyle a_{n}=n^{th}$ term or general term
$\displaystyle l$=last term
$\displaystyle d$=common difference
$\displaystyle n$=number of terms
$\displaystyle S_{n}$=sum of n terms of an A.P.
Let $\displaystyle a$ be the first term and $\displaystyle d$ be the common difference of an A.P., then the A.P. is $\displaystyle a,a+d,a+2d,a+3d...$
$\displaystyle a_{n}=\displaystyle a+(n-1)d$
$\displaystyle l=a+(n-1)d$
$\displaystyle S_{n}=\frac{n}{2}[2a+(n-1)d]$ OR
$\displaystyle S_{n}=\frac{n}{2}[a+l]$
Note:
i) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
ii) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
iii) If each term of an A.P. is multiplied by a constant, the resulting sequence is also an A.P.
iv) If each term of an A.P. is divided by a non-zero constant, the resulting sequence is also an A.P.
v) If the sum of n terms of an A.P. is denoted by $\displaystyle S_{n}$, then its common difference =$\displaystyle S_{2}-2S_{1}$.
vi) The sum of the terms equidistant from the beginning and the end of an A.P. is always the same and equals to the sum of the first and the last terms.
Arithmetic Mean (A.M.)
Given two numbers $\displaystyle a$ and $\displaystyle b$. We can insert a number $\displaystyle A$ between them so that $\displaystyle a,A,b$ is an A.P.
Number $\displaystyle A$ is called arithmetic mean of numbers $\displaystyle a$ and $\displaystyle b$.
Number $\displaystyle A$ is called arithmetic mean of numbers $\displaystyle a$ and $\displaystyle b$.
$\displaystyle A-a=b-A$
⇒ $\displaystyle A=\frac{a+b}{2}$
Similarly, we can insert many numbers between two given numbers $\displaystyle a$ and $\displaystyle b$ such that the resulting sequence is an A.P.
Let $\displaystyle A_{1},A_{2},A_{3},A_{4},...A_{n}$ be $\displaystyle n$ numbers between $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle a,A_{1},A_{2},A_{3},A_{4},...A_{n},b$ is an A.P.
Here, $\displaystyle b$ is $\displaystyle (n+2)^{th}$ term
$\displaystyle b=a+(n+2-1)d=a+(n+1)d$
⇒ $\displaystyle d=\frac{b-a}{n+1}$
Thus, n arithmetic means between a and b are,
$\displaystyle A_{1}=a+d=a+\frac{b-a}{n+1}$
$\displaystyle A_{2}=a+2d=a+\frac{2(b-a)}{n+1}$
$\displaystyle A_{3}=a+3d=a+\frac{3(b-a)}{n+1}$
Similarly, $\displaystyle A_{n}=a+nd=a+\frac{n(b-a)}{n+1}$
a) three numbers are taken as $\displaystyle a-d,a,a+d$
b) four numbers are taken as $\displaystyle a-3d,a-d,a+d,a+3d$
c) five numbers are taken as $\displaystyle a-2d,a-d,a,a+d,a+2d$
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