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Homogeneous Function

Let n ∈ N, Sn...

Question

Let $n \in N, S_{n} = 3 n \sum r = 0^{3 n} C_{r}$ and $T_{n} = n \sum r = 0^{3 n} C_{3 r},$ then $9 | S_{n} - 3 T_{n} |$ is equal to

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Solution

$S_{n} =^{3 n} C_{0} +^{3 n} C_{1} + . . . +^{3 n} C_{3 n} = 2^{3 n}$
$T_{n} =^{3 n} C_{0} +^{3 n} C_{3} + . . . +^{3 n} C_{3 n}$
${(1 + x)}^{3 n} = (^{3 n} C_{0} +^{3 n} C_{3} x^{3} + . . .)$
$+ x (^{3 n} C_{1} +^{3 n} C_{4} x 3 + . . .) + x^{2} (^{3 n} C_{2} +^{3 n} C_{2} x^{5} + . . . .)$
Put $x = 1, ω, ω^{2}$ , we get
$\Rightarrow 2^{3 n} + (- ω^{2})^{3 n} + (- ω)^{3 n} = 3 [^{3 n} C_{0} +^{3 n} C_{3} + . . . .]$
$\Rightarrow 2^{3 n} + (- 1)^{3 n} [ω^{6 n} + ω^{3 n}] = 3 [^{3 n} C_{0} +^{3 n} C_{3} + . . . .]$
$= 3 T_{n}$
$\Rightarrow | S_{n} - 3 T_{n} | = 2$

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