If $a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .$
$b =^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} +^{n} C_{7} +^{n} C_{8} + . . .$
$c =^{n} C_{1} -^{n} C_{2} +^{n} C_{4} -^{n} C_{5} +^{n} C_{7} -^{n} C_{8} + . . .$ ,
then the value of ${(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}$ is

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Solution

Let $ω$ be the cube root of unity.
$(1 + ω)^{n} =^{n} C_{0} +^{n} C_{1} ω +^{n} C_{2} ω^{2} + . . . .$
Since, $ω^{3 n} = 1, ω^{3 n + 1} = ω, ω^{3 n + 2} = ω^{2}, n = 0, 1, 2, . . .$
$\Rightarrow (1 + ω)^{n} = (^{n} C_{0} +^{n} C_{3} + . . .) + (^{n} C_{1} +^{n} C_{4} + . . .) (\frac{- 1 + \sqrt{3} i}{2}) + (^{n} C_{2} +^{n} C_{5} + . . .) (\frac{- 1 - \sqrt{3} i}{2})$
$= (^{n} C_{0} +^{n} C_{3} + . . .) - \frac{1}{2} (^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} + . . . .) + \frac{i \sqrt{3}}{2} (^{n} C_{1} -^{n} C_{2} +^{n} C_{4} -^{n} C_{5} + . . . .)$

$\Rightarrow (1 + ω)^{n} = (a - \frac{b}{2}) + i \frac{\sqrt{3} c}{2}$
Taking modulus both the side, we get
$| (- ω^{2})^{n} | = {(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4} [∵ 1 + ω + ω^{2} = 0]$
$\Rightarrow {(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4} = 1$

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Similar questions

Q. If

a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .

b =^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} +^{n} C_{7} +^{n} C_{8} + . . .

c =^{n} C_{1} -^{n} C_{2} +^{n} C_{4} -^{n} C_{5} +^{n} C_{7} -^{n} C_{8} + . . .

,
then the value of

{(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}

Which of the following is/are true?

(1) $(1 + i)^{n}$ = $2^{\frac{n}{2}}$ $(s i n \frac{n π}{4} + i c o s \frac{n π}{4})$

If $(1 + i)^{n}$ = $^{n} C_{0}$ + $^{n} C_{1}$ i - $^{n} C_{2}$ - $^{n} C_{3}$ i + $^{n} C_{4}$ ...............

(2) $^{n} C_{0}$ - $^{n} C_{2}$ + $^{n} C_{4}$ - $^{n} C_{6}$ ................ = $2^{\frac{n}{2}}$ $s i n \frac{n π}{4}$

(3) $^{n} C_{1}$ - $^{n} C_{3}$ + $^{n} C_{5}$ - $^{n} C_{7}$ ....................= $2^{\frac{n}{2}} c o s \frac{n π}{4}$

If $S_{1}$ = $^{n} C_{0}$ + $^{n} C_{1}$ + $^{n} C_{2}$ .................. $^{n} C_{n}$ and $S_{2}$ = $^{n} C_{0}$ - $^{n} C_{1}$ + $^{n} C_{2}$ .................... $^{n} C_{n}$

Find $^{n} C_{1}$ + $^{n} C_{3}$ + $^{n} C_{5}$ ..............

Q. If

^{n} C_{4} =^{n} C_{5}

, then the value of

^{n} C_{2}

Q. We know that if

^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}

be binomial coefficients then

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + \dots + C_{n} x^{n}

. Various relations among binomial coefficients can be derived by putting

x = 1, - 1, x = i, x = w, where, i = \sqrt{- 1}, w = - \frac{1}{2} + \frac{i \sqrt{3}}{2}

. Some other identities can be derived by adding and subtracting two such identities. The expression

(a + i b)^{n}

can be evaluated by using De-Moiver's theorem by putting

a = r cos θ, b = r sin θ

.

The value of the expression

(^{n} C_{0} -^{n} C_{2} +^{n} C_{4} -^{n} C_{6} + \dots)^{2} + (^{n} C_{1} -^{n} C_{3} +^{n} C_{5} \dots)^{2}

must be

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If a=nC0+nC3+nC6+... b=nC1+nC2+nC4+nC5+nC7+nC8+... c=nC1−nC2+nC4−nC5+nC7−nC8+..., then the value of (a−b2)2+3c24 is

If $a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .$
$b =^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} +^{n} C_{7} +^{n} C_{8} + . . .$
$c =^{n} C_{1} -^{n} C_{2} +^{n} C_{4} -^{n} C_{5} +^{n} C_{7} -^{n} C_{8} + . . .$ ,
then the value of ${(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}$ is