JEE Main & Advanced Mathematics Sequence & Series Arithmetic Mean

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

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

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