Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^{2}}\right)^{n}, x \neq 0, n \in N$, be 376. Then the coefficient of $x^{4}$ is ____.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Binomial Theorem

Let the sum of the coefficients of the first three terms in the expansion of

(x - \frac{3}{x ^{2}})^{n}, x \neq = 0, n \in N

, be 376. Then the coefficient of

x^{4}

is ____.

Answer: 405

The general term of

(x - \frac{3}{x ^{2}})^{n}

(r n) (- 3)^{r} x^{n - 3 r}

, so the first three coefficients are

1

- 3 n

and

\frac{9 n ( n - 1 )}{2}

. Their sum is

1 - 3 n + \frac{9 n ( n - 1 )}{2} = 376

, which reduces to

3 n^{2} - 5 n - 250 = 0

, i.e.

(n - 10) (3 n + 25) = 0

; the natural-number root is

n = 10

. The term in

x^{4}

needs

n - 3 r = 4

, so

r = 2

, and its coefficient is

(2 10) (- 3)^{2} = 45 \times 9 = 405

. The coefficient of

x^{4}

405

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Integer answer