Let the sum of the coefficients of the first three terms in the expansion of (x - 3/x²)ⁿ, x ≠ 0, n ∈

Q: Wait, why do we reject the negative root when solving for $n$?

That is a fantastic question! In the context of the Binomial Theorem, $n$ represents the exponent of the binomial expansion, which defines the number of terms as $n+1$. By definition, $n$ must be a natural number ($n \in \mathbb{N}$). A negative value or a fraction would not make sense in the context of a standard binomial expansion. Always check your domain constraints!

Q: Why can't we just use the sum of coefficients property $P(1)$?

The property that the sum of all coefficients is $P(1)$ (by setting $x=1$) is very powerful, but it applies to the sum of *all* coefficients in the expansion. Here, the problem specifically asks for the sum of only the *first three* coefficients. Using $P(1)$ would give you the sum of all $n+1$ terms, which is not what we need. We must extract them individually using the general term formula.

Q: How do I know if $r=2$ is the correct term for $x^4$?

Great intuition! We don't guess; we calculate. After finding $n=10$, the general term becomes $T_{r+1} = {}^{10}C_r (-3)^r x^{10-3r}$. To find the coefficient of $x^4$, we simply set the exponent of $x$ equal to $4$, so $10 - 3r = 4$. Solving this linear equation gives $3r = 6$, so $r = 2$. It is a precise, logical process, not a guess!

Question

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

Sigma Percentile · Accepted Answer

### The Elegance of the Binomial Expansion

Welcome, fellow traveler on the path of mathematics. Today, we are not just solving a problem; we are peeling back the layers of a beautiful algebraic structure.

The Binomial Theorem is one of the most powerful tools in our arsenal, acting as a bridge between simple arithmetic and the complex world of combinatorics. When you look at an expression like $(x - \frac{3}{x^2})^n$, it might seem daunting at first. But remember, every complex expression is just a collection of simpler parts waiting to be organized.

### Phase 1: The Master Key

Before we dive into the numbers, we must equip ourselves with the right tool. In the world of binomial expansions, the general term formula is our master key.

For any expansion of the form $(a + b)^n$, the general term $T_{r+1}$ is given by:

$$T_{r+1} = {}^{n}C_{r} a^{n-r} b^{r}$$

This formula is the heartbeat of the problem. It tells us exactly what each term looks like without having to write out the entire expansion.

In our specific case, we identify $a = x$ and $b = -\frac{3}{x^2}$. Notice the negative sign attached to the $3$. By keeping that negative sign inside the term $b$, we ensure that our coefficients carry the correct polarity throughout our calculations.

### Phase 2: Extracting the Coefficients

The problem gives us a specific clue: the sum of the coefficients of the first three terms is $376$. To find these, we look at the general term for $r=0, 1, 2$.

For $r=0$, we have the first term:

$$T_{1} = {}^{n}C_{0} (x)^{n} \left( - \frac{3}{x^2} ight)^{0} = 1$$

The coefficient here is simply $1$. For $r=1$, we have:

$$T_{2} = {}^{n}C_{1} (x)^{n-1} \left( - \frac{3}{x^2} ight)^{1} = -3n x^{n-3}$$

The coefficient is $-3n$. Finally, for $r=2$, we have:

$$T_{3} = {}^{n}C_{2} (x)^{n-2} \left( - \frac{3}{x^2} ight)^{2} = \frac{n(n-1)}{2} \cdot 9 x^{n-6}$$

The coefficient is $\frac{9n(n-1)}{2}$. We have successfully extracted the numerical DNA of the first three terms. Now, we assemble them into our equation:

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

### Phase 3: The Algebraic Pivot

Now, we face the quadratic equation. Let us simplify it with care. Multiplying the entire equation by $2$ clears the fraction:

$$2 - 6n + 9n(n-1) = 752$$

Expanding the brackets gives us $2 - 6n + 9n^2 - 9n = 752$. Combining like terms, we arrive at:

$$9n^2 - 15n - 750 = 0$$

Dividing by $3$ makes our lives much easier:

$$3n^2 - 5n - 250 = 0$$

We need to factorize this. We are looking for two numbers that multiply to $3 	imes (-250) = -750$ and add to $-5$. Those numbers are $-30$ and $25$.

Thus, we write $3n^2 - 30n + 25n - 250 = 0$, which factors into $(n - 10)(3n + 25) = 0$. Since $n$ must be a natural number, we reject the negative fraction and confidently declare **$n = 10$**.

### Phase 4: The Final Hunt

With $n=10$ in our possession, the expansion is fully defined. We return to our general term:

$$T_{r+1} = {}^{10}C_{r} (x)^{10-r} \left( - \frac{3}{x^2} ight)^{r}$$

Simplifying this, we get:

$$T_{r+1} = {}^{10}C_{r} (-3)^{r} x^{10-3r}$$

We want the coefficient of $x^4$. This means the exponent of $x$ must be $4$. So, we set $10 - 3r = 4$, which leads us to $3r = 6$, or $r = 2$.

The term we need is the third term ($r=2$). Substituting $r=2$ into our coefficient expression ${}^{10}C_{r} (-3)^{r}$, we get:

$${}^{10}C_{2} (-3)^{2} = 45 	imes 9 = 405$$

### Conclusion

Look at what you have achieved. You started with a cryptic expression and, through logical deduction and algebraic rigor, arrived at the exact coefficient.

The beauty of this problem lies not just in the final answer of **$405$**, but in the journey of translating the language of the problem into the language of mathematics. Keep this clarity of thought, and no problem will ever be too complex for you.

Animated Solution for Mathematics - 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 ______.

Enter Numerical Value:

Visualized Solution

The Binomial Expansion Problem

The General Term Formula

Applying the General Term

Extracting the First Three Coefficients

Setting Up the Sum Equation

Simplifying to a Quadratic Equation

Solving for $n$

Updating the General Term

Simplifying the Powers of $x$

Finding $r$ for $x^{4}$

Calculating the Final Coefficient

Summary and Final Answer

The Sigma Insight: Binomial Expansion for Positive Integral Index

The Elegance of the Binomial Expansion

Phase 1

Phase 2

Phase 3

Phase 4

Conclusion

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