Question

Prove that for the terms of an arithmetic progression the equality \[{a_{p\;}} + {\text{ }}{a_{m\;}} = {\text{ }}{a_{p + k}} + {\text{ }}{a_{m - k}}\]

Answer 1

Hint:Here we use the formula for \[{n^{th}}\] term of Arithmetic progression which helps us to define the terms of an A.P with the help of the first term and the common difference between the terms and write all the terms in the question with the help of the formula and solve for LHS and RHS separately.
* The \[{n^{th}}\] term of A.P is given by \[{T_n} = {\text{ }}a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right)d\] where \[a\] is the first term of A.P and \[d\]is the common difference between the terms of an A.P.

Complete step-by-step answer:
First we write all the terms given in the question in form of the formula \[{T_n} = {\text{ }}a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right)d\]
Using the formula for \[{p^{th}}\] term \[{a_p} = {\text{ }}a{\text{ }} + {\text{ }}\left( {p{\text{ }} - 1} \right)d\]
Using the formula for \[{m^{th}}\] term \[{a_m} = {\text{ }}a{\text{ }} + {\text{ }}\left( {m - 1} \right)d\]
Using the formula for \[{(p + k)^{th}}\]term \[{a_{p + k}} = {\text{ }}a{\text{ }} + {\text{ }}\left( {p{\text{ + k}} - 1} \right)d\]
 Using the formula for\[{(m - k)^{th}}\] term \[{a_{m - k}}\; = a{\text{ }} + {\text{ }}\left( {m - {\text{ }}k{\text{ }} - 1} \right)d\]
Solving the LHS of the equation by substituting the values
\[{a_{p\;}} + {\text{ }}{a_m} = a{\text{ }} + \left( {p{\text{ }} - 1} \right)d{\text{ }} + {\text{ }}a{\text{ }} + \left( {m - 1} \right)d\]
                 \[\begin{array}{*{20}{l}}
  { = a + {\text{ }}pd{\text{ }}-d{\text{ }} + a{\text{ }} + md{\text{ }}-d} \\
  { = 2a{\text{ }} + pd{\text{ }} + md{\text{ }} - 2d}
\end{array}\]
Now solving the RHS of the equation by substituting the values
\[{a_{p + k}} + {\text{ }}{a_{m - k}} = a{\text{ }} + {\text{ }}\left( {p{\text{ }} + {\text{ }}k{\text{ }} - 1} \right)d{\text{ }} + {\text{ }}a{\text{ }} + \left( {m{\text{ }}-{\text{ }}k{\text{ }} - 1} \right)d\]
                        \[\begin{array}{*{20}{l}}
  { = a + pd{\text{ }} + kd{\text{ }}-d{\text{ }} + a{\text{ }} + md{\text{ }}-kd{\text{ }}-d} \\
  { = 2a{\text{ }} + pd{\text{ }} + md{\text{ }}-2d}
\end{array}\]
On comparing the solution of LHS of the equation and RHS of the equation we get the same value.
Therefore, \[{a_{p\;}} + {\text{ }}{a_{m\;}} = {\text{ }}{a_{p + k}} + {\text{ }}{a_{m - k}}\]
Additional information:
An arithmetic progression is a sequence of terms in which difference of any two consecutive terms is always the same.
Common difference is given by \[d = {a_{n + 1}} - {a_n}\]
The sum of an AP is given by \[S = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]\]
* Similar to arithmetic progression there are other sequences like Geometric progression denoted as G.P and harmonic progression denoted as H.P.

Note:Students are likely to make mistakes when opening the brackets for the term formula, so calculations must be done very carefully and step by step. Keep in mind that an arithmetic progression series common difference can be zero, negative and positive.