# JEE Main & Advanced Mathematics Arithmetic Progressions Arithmetic Mean

## Arithmetic Mean

Category : JEE Main & Advanced

If $a,A,b$ are in A.P., then A is called A.M. between $a$ and $b$.

(1) If $a,\,{{A}_{1}},\,{{A}_{2}},\,{{A}_{3}},.....,\,{{A}_{n}},\,b$ are in A.P., then ${{A}_{1}},\,{{A}_{2}},\,{{A}_{3}},\,......,\,{{A}_{n}}$ are called $n$ A.M.?s between $a$ and $b$.

(2) Insertion of arithmetic means

(i) Single A.M. between $a$ and $b$ : If $a$ and $b$ are two real numbers then single A.M. between $a$ and $b$$=\frac{a+b}{2}$

(ii) n A.M.?s between a and b : If ${{A}_{1}},\,{{A}_{2}},\,{{A}_{3}},\,.......,\,{{A}_{n}}$ are n A.M.?s between $a$ and $b$, then

${{A}_{1}}=a+d=a+\frac{b-a}{n+1}$, ${{A}_{2}}=a+2d=a+2\frac{b-a}{n+1}$,

${{A}_{3}}=a+3d=a+3\frac{b-a}{n+1}$, ??., ${{A}_{n}}=a+nd=a+n\frac{b-a}{n+1}$.

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