For AP show that $a_{p} + a_{p + 2 q} = 2 a_{p + q}$

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Solution

L.H.S $= a_{p} + a_{p + 2 q}$

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

$= a + d p - d + a + d p + 2 d q - d$

$= 2 a + 2 d p - 2 d + 2 d q = 2 (a + d p - d + d q) . . . . (i)$

R.H.S
$= 2 a_{p + q}$

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

$= [(a + d p + d q - d)] 2...... (i i)$

Hence from (i) and (ii) ,
L.H.S= R.H.S

HENCE PROVED

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