In the binomial expansion of $ \left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n} $, the ratio of the $ 7^{\text {th }} $ term from the beginning to the $ 7^{\text {th }} $ term from the end is $ 1: 6 $, then the value of $ n $ is

Question

In the binomial expansion of $ \left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n} $, the ratio of the $ 7^{\text {th }} $ term from the beginning to the $ 7^{\text {th }} $ term from the end is $ 1: 6 $, then the value of $ n $ is, $ 13 $, $ 16 $, $ 9 $, $ 23 $

MARKS App · Accepted Answer

7 t h term from the beginning is C 6 n 2 n - 6 3 1 3 6 3 7 t h term from the end is C 6 n 1 3 n - 6 3 2 6 3 ⇒ n C 6 2 n 3 - 2 1 3 2 n C 6 2 2 1 3 n 3 - 2 = 1 6 ⇒ 2 n 3 - 4 1 3 4 - n 3 = 1 6 ⇒ 2 × 3 n 3 - 4 = 2 × 3 - 1 ⇒ n = 9

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