Arithmetic Mean (A.M.)

Mathematics Class XI Chapter 6: Sequence and Series Arithmetic Mean (A.M.)

When three numbers $a, m$ and $b$ are in A.P., then $m$ is average of $a$ and $b$ .

Then $m=\dfrac{a+b}{2}$

In general if $a_1,a_2,...,a_n$ are in A.P, their Arithmetic Mean A.M is defined as $A.M=\dfrac{a_1+a_2+...,+a_n}{n}=\dfrac{1}{n}\sum_{k=1}^{n}a_k$

Note -:

The terms $a_2,a_3,...a_{n-1}$ are called the Arithmetic Means between $a_1$ and $a_n$ .

Some Facts about an A.P

(i) If a fixed number is added to (subtracted from) each term of a given AP, then the resulting sequence is also an AP and it has the same common difference as that of the given AP.

(ii) If each term of an AP is multiplied by a fixed constant (or divided by a non-zero fixed constant), then the resulting sequence is also an AP.

(iii) If $a_1,a_2,...$ and $b_1,b_2,…$ are two Arithmetic Progressions, then $a_1+b_1,\ a_2+b_2,...$ is also an AP.

(iv) If we have to take three terms in AP whose sum is known, it is convenient to take them as $a-d,\ a,\ a+d$ . In general, if we take $(2r+1)$ terms in AP, we take them as $a-rd,\ a-(r-1)d,..., a-d,\ a,\ a+d,...a+rd$ .

(v) The $n$ numbers $A_1, A_2,...A_n$ are said to be Arithmetic Means between $a$ and $b$ , if $a,A_1, A_2,...A_n,b$ is an AP.

Here $a$ is the first term and $b$ is the $(n+2)^{th}$ term of the AP.

If $d$ is the common difference of this AP, then $b=a+(n+2-1)d=a+(n+1)d\Rightarrow d=\dfrac{b-a}{n+1}$

Thus, $A_1 = a+\dfrac{b-a}{n+1}$

$A_2 = a+\dfrac{2(b-a)}{n+1}$

$A_n = a+\dfrac{n(b-a)}{n+1}$