Question

If $\sum _{r=0}^n{C}_r{x}^r$ , then the value of $a{C}_0-(a+d)C_1+(a-2d)C_2-(a-3d)C_3+\cdots +{(-1)}^n(a-nd)C_n$ , equals

• A. $0$
• B. $(2a-n)2^n{(-1)}^n$
• C. ${(-1)}^n(a+nd)2^n$
• D. None of these

Answer 1

The correct option is A $0$

Simplifying, we get
$=a({}^n{C}_0-{}^n{C}_1+{}^n{C}_2+...(-1{)}^n{}^n{C}_n)-d({}^n{C}_1-2{}^n{C}_2+3{}^n{C}_3+...(-1{)}^{n}dn{}^n{C}_n)$
$=\left[a\right(1-x{)}^n+d\frac{d(1-x{)}^n}{dx}]{|}_{x=1}$
$=0$
Hence answer is $0.$