Show that the sum of $(m + n)^{t h}$ and $(m - n)^{t h}$ term of an $A . P .$ ie equal to twice the $m^{t h}$ terms.

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Solution

To prove $2 a_{m} = a_{m + n} + a_{m - n}$

$a_{n} = a + (n - 1) d$

Similarly,

$a_{m + n} = a + (m + n - 1) d$

and $a_{m - n} = a + (m - n - 1) d$

$a_{m + n} + a_{m - n}$

$= a + (m + n - 1) d + a + (m - n - 1) d$

$= 2 a + m d + n d - d + a + m d - n d - d$

$= 2 a + 2 m d - 2 d$

$= 2 [a + (m - 1) d]$

$= 2 a_{m}$

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