Question

If $C _0, C _1 \ldots \ldots, C _{2012}$ are binomial coefficients in the expansion of $(1+ x )^{2012}$ and $a _0, a _1 \ldots \ldots a _{2012}$ are real numbers in arithmetic progression then value of $a _0 C _0- a _1 C _1+ a _2 C _2- a _3 C _3+\ldots . .+ a _{2012} C _{2012}$, is

• A. 1
• B. 2012
• C. -1
• D. 0

Answer 1

Answer: (D)

$ a _0 C _0- a _1 C _1+ a _2 C _2- a _2 C _3+\ldots \ldots+ a _{2012} C _{2012} $
$= a _0\left( C _0- C _1+ C _2+\ldots \ldots+ C _{2012}\right)- d \left( C _1-2 C _2+3 C _3-4 C _4+\ldots \ldots{ }^{-2012} C _{2012}\right) $
$= a _0(0)- d \left( C _1-2 C _2+3 C _3-4 C _4+\ldots \ldots .{ }^{-2012} C _{2012}\right) $
$\text { Now }(1+ x )^{2012}= C _0+ C _1 x + C _2 x ^2+\ldots \ldots+ C _{2012} x ^{2012} $
$\text { Diff. and put } x =-1 $
$0= C _1-2 C _2+3 C _3-4 C _4+\ldots \ldots \ldots .{ }^{-2012} C _{2012} $