If the sum of the coefficients of and in the expansion of , , is zero, then the value of n is ________. [2026]

Question

If the sum of the coefficients of $x^{7}$ and $x^{14}$ in the expansion of ${(\frac{1}{x^{3}} - x^{4})}^{n}$ , $x \neq 0$ , is zero, then the value of n is ________. [2026]

Smart Achievers · Accepted Answer

(21)

We have, ${(\frac{1}{x^{3}} - x^{4})}^{n}$

General term is given by

$T_{r + 1} = {}^{n}C_{r} {(\frac{1}{x^{3}})}^{n - r} {(- x^{4})}^{r}$

$= {}^{n}C_{r} x^{- 3 n + 3 r + 4 r} {(- 1)}^{r} = {}^{n}C_{r} x^{- 3 n + 7 r} {(- 1)}^{r}$

$∴ - 3 n + 7 r = 7 \Rightarrow 7 r = 7 + 3 n$

$\Rightarrow r \frac{7 + 3 n}{7} \Rightarrow r = 1 + \frac{3 n}{7}$

$∴ - 3 n + 7 r = 14 \Rightarrow 7 r = 14 + 3 n$

$\Rightarrow r = 2 + \frac{3 n}{7}$

Since, Coefficient of $x^{7}$ + Coefficient of $x^{14} = 0$

$∴ {}^{n}C_{(1 + \frac{3 n}{7})} {(- 1)}^{1 + \frac{3 n}{7}} + {}^{n}C_{(2 + \frac{3 n}{7})} {(- 1)}^{2 + \frac{3 n}{7}} = 0$

$\Rightarrow {(- 1)}^{1 + \frac{3 n}{7}} [{}^{n}C_{(1 + \frac{3 n}{7})} - {}^{n}C_{(2 + \frac{3 n}{7})}] = 0$

$\Rightarrow {}^{n}C_{(1 + \frac{3 n}{7})} = {}^{n}C_{(2 + \frac{3 n}{7})}$

$\Rightarrow 1 + \frac{3 n}{7} + 2 + \frac{3 n}{7} = n \Rightarrow \frac{7 n - 6 n}{7} = 3$

$n = 21$

Solution

Q.
If the sum of the coefficients of $x^{7}$ and $x^{14}$ in the expansion of ${(\frac{1}{x^{3}} - x^{4})}^{n}$ , $x \neq 0$ , is zero, then the value of n is ________. [2026]

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Solution

Q. If the sum of the coefficients of x7 and x14 in the expansion of (1x3–x4)n, x≠0, is zero, then the value of n is ________. [2026]

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Q.
If the sum of the coefficients of $x^{7}$ and $x^{14}$ in the expansion of ${(\frac{1}{x^{3}} - x^{4})}^{n}$ , $x \neq 0$ , is zero, then the value of n is ________. [2026]