The sum of the real values of x for which the middle terms in the binomial expansion of equals 5670 is

Engineering

Mathematics

Application of Binomial Theorem and Summation Of Series

Question

The sum of the real values of x for which the middle terms in the binomial expansion of ${(\frac{x^{3}}{3} + \frac{3}{x})}^{8}$ equals 5670 is

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5th term will be the middle term.

$t_{4 + 1} =^{8} C_{4} {(\frac{x^{3}}{3})}^{4} {(\frac{3}{x})}^{4} = 5670$

= ⁸C₄.x⁸ = 5670

$= \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2} x^{8} = 5670$

$= x^{8} = \frac{567}{7} = 81$

= x⁸ – 81 = 0

⇒ Real value of $x = \pm \sqrt{3}$