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Sum of Coefficients of All Terms

If 1+ x + x...

Question

If $(1 + x + x^{2}) n = a 0 + a^{1} x + a^{2} x + a^{3} x^{3} + . . . . . . . a (2^{n}) x (2^{n})$ then prove that $(a^{0} + a^{3} + a^{6} + . . . . . . . . .) = (a^{1} + a^{4} + a^{7}) = (a^{2} + a^{5} + a^{8}) = 3^{(} n - 1),$

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Solution

put = $ω a n d n o t e t h a t 1 + ω + ω^{2} = 0$
where $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2} a n d ω^{2} = - \frac{1}{2} - i \frac{\sqrt{3}}{2}$
$Δ = (a_{0} + a_{3} + a_{6} + . . . . . .) + ω (a_{1} + a_{4} + . . . . . .) + ω (a_{2} + a_{3} . . . . . .)$
or $Δ = A ω + B ω^{2} + C = 0$
equating real and imaginary part of both sides
$A - \frac{B}{2} - \frac{C}{2} = 0 \frac{\sqrt{3}}{2} (B - C) = 0$
B = C and hence A = B
A \, = \, B \, = \., C
Again putting x = 1 in the given relation . we get 3^ n = sum of the all the condition = A + B + C

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Q. If the expansion of

(1 + x + x^{2})^{n}

a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{r} x^{r} + . . . . . + a_{2 n} x^{2 n}

a_{0} + a_{3} + a_{6} + . . . . . = a_{1} + a_{4} + a_{7} + . . . . . = a_{2} + a_{5} + a_{8} + . . . . = 3^{n - 1}

(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{2 n} x^{2 n}

E_{1} = a_{0} + a_{3} + a_{6} + . . .

E_{2} = a_{1} + a_{4} + a_{7} + . . .

E_{3} = a_{2} + a_{5} + a_{8} + . . .

(1 + x + x^{2})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . + a_{2 n} x^{2 n}

\begin{matrix} A = a_{0} + a_{3} + a_{6} + . . . + a_{3 k} + & . . . B = a_{1} + a_{4} + a_{7} + . . . + a_{3 k + 1} + & . . . C = a_{2} + a_{5} + a_{8} + . . . + a_{3 k + 2} + & . . . \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭ \forall k = 0, 1, . . . n

then which of following is/are true?

a_{0}, a_{1}, a_{2}, . . . . .

be the coefficients in the expansion of

(1 + x + x^{2})^{n}

in ascending powers of x, then prove that :

(a_{0} + a_{3} + a_{6} + . . .) = (a_{1} + a_{4} + a_{7} + . . . .) = = (a_{2} + a_{5} + a_{8} + . . .) = 3^{n - 1}

{(1 + x + x^{2})}^{n} = 2 n \sum r = 0 a_{r} x^{r} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a^{2 n} x^{2 n}

P = a_{0} + a_{3} + a_{6} + . . .

Q = a_{1} + a_{4} + a_{7} + . . .

R = a_{2} + a_{5} + a_{8} + . . .

then the set of values of

P, Q, R

are respectively equals

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