Question

If $C_0,C_1\dots \dots ,C_{2012}$ are binomial coefficients in the expansion of $(1+x{)}^{2012}$ and $a_0,a_1\dots \dots a_{2012}$ are real numbers in arithmetic progression then value of $a_0{C}_0-a_1{C}_1+a_2{C}_2-a_3{C}_3+\dots ..+a_{2012}{C}_{2012}$, is

• A. 1
• B. 2012
• C. -1
• D. 0

Answer 1

Answer: (D)

$a_0{C}_0-a_1{C}_1+a_2{C}_2-a_2{C}_3+\dots \dots +a_{2012}{C}_{2012}$
$=a_0\left(C_0-C_1+C_2+\dots \dots +C_{2012}\right)-d\left(C_1-2C_2+3C_3-4C_4+\dots \dots {}^{-2012}{C}_{2012}\right)$
$=a_0(0)-d\left(C_1-2C_2+3C_3-4C_4+\dots \dots .{}^{-2012}{C}_{2012}\right)$
$\text{ Now }(1+x{)}^{2012}=C_0+C_{1}x+C_2{x}^2+\dots \dots +C_{2012}{x}^{2012}$
$\text{ Diff. and put }x=-1$
$0=C_1-2C_2+3C_3-4C_4+\dots \dots \dots .{}^{-2012}{C}_{2012}$