Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

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}

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