Progressions
3. Arithmetic Progression
3.3. Properties of A.P.
- If \({A_1},{\rm{ }}{A_2},{\rm{ }}{A_3}, \ldots \ldots {A_n}\) are in A.P. having common difference \(‘d’\), then \({A_1} \pm x,\,\,{A_2} \pm x,\,\,......\) are also in A.P. having the same common difference \(‘d’\).
- If \({A_1},{\rm{ }}{A_2},{\rm{ }}{A_3},{\rm{ }} \ldots \ldots \ldots \) are in A.P. having common difference \(d\), then \(k{A_1},{\rm{ }}k{A_2},{\rm{ }}k{A_3}, \ldots ..\) are also in A.P. having common difference equal to \(‘kd’\).
- If \({n^{th}}\) term of a series is linear expression in \(n\) which means that \({T_n} = {\rm{ }}an{\rm{ }} + {\rm{ }}b\), then \({T_n}\) definitely represents \({n^{th}}\) term of an \(A.P.\) If \({T_n}\) is not a linear expression in \(n\), then it is definitely not an \(A.P.\)
- If \({A_1},{\rm{ }}{A_2},{\rm{ }}{A_3}, \ldots ..,{\rm{ }}{A_n}\) are \(n\) numbers, then the arithmetic mean \(A\), of these numbers is given by
\(A = \frac{1}{n}({A_1} + {A_2} + {A_3} + \,\,......\,\, + {A_n})\) - In an AP, sum of two terms which are equidistant from the two ends of the series, is constant and is equal to double of the Arithmetic Mean. If \({A_1},{\rm{ }}{A_2},{\rm{ }}{A_3}, \ldots ..,{\rm{ }}{A_{10}}\) are 10 numbers in \(AP\), then
\({A_1} + {\rm{ }}{A_{10}} = {\rm{ }}{A_2} + {\rm{ }}{A_9} = {\rm{ }}{A_3} + {\rm{ }}{A_8} \ldots {\rm{ }} = {\rm{ }}2\left( {AM} \right)\) - If there are \(2n + 1\) terms in an \(AP\), then ratio of sum of odd terms to the sum of even terms will be: \(\frac{{n + 1}}{n}\).