JEE Main & Advanced Mathematics Sequence & Series Properties of A.P.

(1) If \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}}.....\] are in A.P. whose common difference is \[d,\] then for fixed non-zero number \[k\in R\].

(i) \[{{a}_{1}}\pm k,\,{{a}_{2}}\pm k,\,{{a}_{3}}\pm k,.....\] will be in A.P., whose common difference will be \[d\].

(ii) \[k{{a}_{1}},\,k{{a}_{2}},\,k{{a}_{3}}....\] will be in A.P. with common difference \[=kd\].

(iii) \[\frac{{{a}_{1}}}{k},\,\frac{{{a}_{2}}}{k},\,\frac{{{a}_{3}}}{k}......\] will be in A.P. with common difference \[=d/k\].

(2) The sum of terms of an A.P. equidistant from the beginning and the end is constant and is equal to sum of first and last term. i.e. \[{{a}_{1}}+{{a}_{n}}={{a}_{2}}+{{a}_{n-1}}={{a}_{3}}+{{a}_{n-2}}=....\]

(3) If number of terms of any A.P. is odd, then sum of the terms is equal to product of middle term and number of terms.

(4) If number of terms of any A.P. is even then A.M. of middle two terms is A.M. of first and last term.

(5) If the number of terms of an A.P. is odd then its middle term is A.M. of first and last term.

(6) If \[{{a}_{1}},\,{{a}_{2}},\,......{{a}_{n}}\] and \[{{b}_{1}},\,{{b}_{2}},\,......{{b}_{n}}\] are the two A.P.'s. Then \[{{a}_{1}}\pm {{b}_{1}},\,{{a}_{2}}\pm {{b}_{2}},\,......{{a}_{n}}\pm {{b}_{n}}\] are also A.P.'s with common difference \[{{d}_{1}}\ne {{d}_{2}}\], where \[{{d}_{1}}\] and \[{{d}_{2}}\] are the common difference of the given A.P.'s.

(7) Three numbers \[a,\,\,b,\,\,\,c\] are in A.P. iff \[2b=a+c\].

(8) If \[{{T}_{n}},\,{{T}_{n+1}}\] and \[{{T}_{n+2}}\] are three consecutive terms of an A.P., then \[2{{T}_{n+1}}={{T}_{n}}+{{T}_{n+2}}\].

(9) If the terms of an A.P. are chosen at regular intervals, then they form an A.P.