Let α be the constant term in the binomial expansion of (√x - 6/x^(3/2))ⁿ, n ≤ 15. If the sum of the

Q: Wait, why does setting $x=1$ give the sum of coefficients? Isn't that only for polynomials?

That is a brilliant observation! You are right that this is a standard trick for polynomials. However, in a binomial expansion like $(\sqrt{x} - \frac{6}{x^{\frac{3}{2}}})^n$, even though it involves fractional powers, the expansion results in a series of terms with coefficients. When we set $x=1$, we are essentially evaluating the function at $x=1$. Since $1$ raised to any power is $1$, the variables disappear, and we are left with the sum of the numerical coefficients. It works perfectly here!

Q: How do I know which $n$ to pick if there are multiple possibilities for $n=4r$?

Great question. We found that $n$ must be a multiple of $4$ (i.e., $4, 8, 12$). We don't just guess; we test! We have a rigid constraint: $(-5)^n - \alpha = 649$. By calculating $\alpha$ for each possible $n$ and plugging it into this equation, we find that only $n=4$ satisfies the equality. Always let the constraints guide your search.

Q: Why did we get a negative coefficient for $x^{-n}$? Is that allowed?

Absolutely! Coefficients in binomial expansions can be negative, positive, or even zero. The sign is determined by the term $(-\frac{6}{x^{\frac{3}{2}}})^r$. If $r$ is odd, the term will be negative. Don't let the negative sign intimidate you; it is just part of the mathematical landscape. Trust your algebra!

Question

Let $\alpha$ be the constant term in the binomial expansion of $(\sqrt{x} - \frac{6}{x^{\frac{3}{2}}})^n, n \le 15$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda\alpha$, then $\lambda$ is equal to ————————.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with analyzing the binomial expression $(\sqrt{x} - \frac{6}{x^{\frac{3}{2}}})^n$ under the constraint $n \le 15$. Our goal is to determine the constant term $\alpha$, the coefficient of $x^{-n}$, and the parameter $\lambda$.

To find the sum of all coefficients in any binomial expansion, we apply the **Golden Rule**: substitute the variable $x$ with $1$.

Setting $x=1$ in our expression, we obtain:
$$(1 - 6)^n = (-5)^n$$
This value represents the total sum of all coefficients in the expansion.

### Decoding the Constant Term

The problem states that the sum of the coefficients of the remaining terms (excluding the constant term $\alpha$) is $649$. We can express this relationship as:
$$(-5)^n - \alpha = 649$$

To find $\alpha$, we examine the general term $T_{r+1}$ of the expansion:
$$T_{r+1} = \binom{n}{r} (x^{\frac{1}{2}})^{n-r} \left(-6x^{-\frac{3}{2}}ight)^r$$

Simplifying the powers of $x$, we get:
$$T_{r+1} = \binom{n}{r} (-6)^r x^{\frac{n-r}{2} - \frac{3r}{2}} = \binom{n}{r} (-6)^r x^{\frac{n-4r}{2}}$$

For a term to be a constant, the exponent of $x$ must be zero:
$$\frac{n-4r}{2} = 0 \Rightarrow n = 4r$$

Given $n \le 15$, the possible values for $n$ are $4, 8,$ or $12$.

### The Moment of Truth

We test our candidates for $n$. If $n=4$, then $r=1$. The constant term $\alpha$ is:
$$\alpha = \binom{4}{1} (-6)^1 = 4 	imes (-6) = -24$$

Substituting this into our sum equation:
$$(-5)^4 - (-24) = 625 + 24 = 649$$
The result matches perfectly. Thus, we have identified **$n=4$** and **$\alpha = -24$**.

### The Final Pursuit of $\lambda$

We now find the coefficient of $x^{-n}$, which is the coefficient of $x^{-4}$ since $n=4$. We set the general exponent equal to $-4$:
$$\frac{4-4r}{2} = -4 \Rightarrow 4-4r = -8 \Rightarrow 4r = 12 \Rightarrow r = 3$$

Calculating the coefficient for $r=3$:
$$\binom{4}{3} (-6)^3 = 4 	imes (-216) = -864$$

The problem defines this coefficient as $\lambda\alpha$. Therefore:
$$-864 = \lambda(-24)$$
$$\lambda = \frac{-864}{-24} = 36$$

Through logical deduction, we have determined that **$\lambda = 36$**.

Enter Numerical Value:

Visualized Solution

The Binomial Expression

Sum of All Coefficients

Equation for Remaining Terms

General Term of the Expansion

Condition for the Constant Term

Testing $n = 4$

Verifying the Sum Condition

Finding the Term with $x^{- n}$

Calculating the Coefficient of $x^{- 4}$

Solving for $λ$

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

Decoding the Constant Term

The Moment of Truth

The Final Pursuit of $λ$

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Enter Numerical Value:

Visualized Solution

The Binomial Expression

Sum of All Coefficients

Equation for Remaining Terms

General Term of the Expansion

Condition for the Constant Term

Testing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n=4

Verifying the Sum Condition

Finding the Term with .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x−n

Calculating the Coefficient of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x−4

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } λ

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

Decoding the Constant Term

The Moment of Truth

The Final Pursuit of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } λ

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