Key Concept: The Binomial Theorem

For any binomial expansion of the form $(a+b)^n$, the general term (or $(r+1)^{th}$ term), denoted as $T_{r+1}$, is given by: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ where $n$ is a non-negative integer, and $r$ is an integer such that $0 \le r \le n$. The coefficient of the $(r+1)^{th}$ term is $\binom{n}{r}$.

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Step 1: Determine the Coefficients of the First Three Terms

We are given the expansion ${\left( {x - {3 \over {{x^2}}}} \right)^n}$. Here, $a=x$ and $b = -\frac{3}{x^2}$.

• First Term (r=0): To find the first term, we set $r=0$ in the general term formula. $T_1 = \binom{n}{0} x^{n-0} \left(-\frac{3}{x^2}\right)^0 = 1 \cdot x^n \cdot 1 = x^n$ The coefficient of the first term is $\binom{n}{0} = 1$.

• Second Term (r=1): To find the second term, we set $r=1$ in the general term formula. $T_2 = \binom{n}{1} x^{n-1} \left(-\frac{3}{x^2}\right)^1 = n \cdot x^{n-1} \cdot \left(-\frac{3}{x^2}\right) = -3n \cdot x^{n-3}$ The coefficient of the second term is $\binom{n}{1} \cdot (-3) = n \cdot (-3) = -3n$.

• Third Term (r=2): To find the third term, we set $r=2$ in the general term formula. $T_3 = \binom{n}{2} x^{n-2} \left(-\frac{3}{x^2}\right)^2 = \frac{n(n-1)}{2} \cdot x^{n-2} \cdot \frac{9}{x^4} = \frac{9n(n-1)}{2} \cdot x^{n-6}$ The coefficient of the third term is $\binom{n}{2} \cdot (-3)^2 = \frac{n(n-1)}{2} \cdot 9 = \frac{9n(n-1)}{2}$.

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Step 2: Formulate and Solve the Equation for 'n'

We are given that the sum of the coefficients of the first three terms is 376. Sum of coefficients = (Coefficient of $T_1$) + (Coefficient of $T_2$) + (Coefficient of $T_3$) $1 + (-3n) + \frac{9n(n-1)}{2} = 376$ To solve for $n$, we first simplify the equation by multiplying by 2 to eliminate the fraction: $2 - 6n + 9n(n-1) = 376 \times 2$ $2 - 6n + 9n^2 - 9n = 752$ Combine like terms to form a quadratic equation: $9n^2 - 15n + 2 = 752$ $9n^2 - 15n - 750 = 0$ Divide the entire equation by 3 to simplify: $3n^2 - 5n - 250 = 0$ Now, we solve this quadratic equation for $n$ using the quadratic formula $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $n = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-250)}}{2(3)}$ $n = \frac{5 \pm \sqrt{25 + 3000}}{6}$ $n = \frac{5 \pm \sqrt{3025}}{6}$ Since $\sqrt{3025} = 55$: $n = \frac{5 \pm 55}{6}$ This gives us two possible values for $n$: $n_1 = \frac{5 + 55}{6} = \frac{60}{6} = 10$ $n_2 = \frac{5 - 55}{6} = \frac{-50}{6} = -\frac{25}{3}$ Given that $n \in \mathbb{N}$ (n is a natural number), we must reject the negative fractional value. Therefore, $n = 10$.

Tip: Always verify your solutions against the constraints given in the problem statement. Here, $n \in \mathbb{N}$ is crucial.

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Step 3: Write the General Term for the Expansion with the Determined 'n'

Now that we know $n=10$, we can write the general term for the expansion ${\left( {x - {3 \over {{x^2}}}} \right)^{10}}$: $T_{r+1} = \binom{10}{r} x^{10-r} \left(-\frac{3}{x^2}\right)^r$ To find the power of $x$, we combine the $x$ terms: $T_{r+1} = \binom{10}{r} x^{10-r} (-3)^r (x^{-2})^r$ $T_{r+1} = \binom{10}{r} (-3)^r x^{10-r-2r}$ $T_{r+1} = \binom{10}{r} (-3)^r x^{10-3r}$

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Step 4: Find the Value of 'r' for the Coefficient of $x^4$

We need to find the coefficient of $x^4$. This means we need to set the power of $x$ in our general term equal to 4: $10 - 3r = 4$ Subtract 10 from both sides: $-3r = 4 - 10$ $-3r = -6$ Divide by -3: $r = 2$

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Step 5: Calculate the Coefficient of $x^4$

Now that we have $n=10$ and $r=2$, we can substitute these values back into the coefficient part of the general term formula: Required coefficient $= \binom{10}{2} (-3)^2$ First, calculate $\binom{10}{2}$: $\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2!8!} = \frac{10 \times 9}{2 \times 1} = 5 \times 9 = 45$ Next, calculate $(-3)^2$: $(-3)^2 = 9$ Finally, multiply these two results: Required coefficient $= 45 \times 9 = 405$

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Summary and Key Takeaway

This problem demonstrates a classic application of the Binomial Theorem. The core idea is to first use the given information about the sum of initial coefficients to determine the value of 'n' for the binomial expansion. Once 'n' is known, the general term formula helps isolate the power of 'x' and solve for 'r' to find the specific term requested. Careful handling of negative signs and exponents is crucial to avoid errors. This methodical approach ensures all parts of the problem are addressed accurately.