# In the expansion of $(x+a)^n$ if the sum of odd terms is denoted by O and the sum of even term by E.Then prove that (ii) $4OE=(x+a)^{2n}-(x-a)^{2n}$

$[(x+a)^n]^2=(O+E)^2$
$\Rightarrow O^2+E^2+2OE$------(1)
$[(x-a)^n]^2=(O-E)^2$
$\Rightarrow O^2+E^2-2OE$------(2)
Subtracting (2) from (1) we get
$(O^2+E^2+2OE)-(O^2+E^2-2OE)$
$\Rightarrow O^2+E^2+2OE-O^2-E^2+2OE$
$\Rightarrow 4OE$
Hence $4OE=(x+a)^{2n}-(x-a)^{2n}$
Hence proved.