Question

Let f(x) = $(1+x{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$ $x^2$ + ${}^n{C}_2$ $x^3$ ........${}^n{C}_n$ $x^n$.

If f(1) = $S_1$, f(w) = $S_2$ and $f\left(w^2\right)$ = $S_3$, find the value of ${}^n{C}_0$ + ${}^n{C}_3$ + ${}^n{C}_6$ + .......... in terms of $S_1$, $S_2$ and

$S_3$.[W is the complex cube root of unity]

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

f(1) = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$ + ${}^n{C}_3$ + ${}^n{C}_4$ + ${}^n{C}_5$ + ${}^n{C}_6$................

f(w) = ${}^n{C}_0$ + ${}^n{C}_1$ w + ${}^n{C}_2$ $w^2$ + ${}^n{C}_3$ + ${}^n{C}_4$ w + ${}^n{C}_5$$w^2$ + ${}^n{C}_6$..........

$f\left(w^2\right)$ = ${}^n{C}_0$ + ${}^n{C}_1$$w^2$ + ${}^n{C}_2$w + ${}^n{C}_3$ + ${}^n{C}_4$$w^2$ + ${}^n{C}_5$w + ${}^n{C}_6$...........

f(1) + f(w) + $f\left(w^2\right)$ = $3{(}^n{C}_0{+}^n{C}_3{+}^n{C}_6..............)$

⇒ ${}^n{C}_0$ + ${}^n{C}_3$ + ${}^n{C}_6$......... = $\frac{1}{3}$ $(S_1+S_2+S_3)$