1. Introduction: The Binomial Theorem and General Term

To find the constant term in the expansion of a binomial expression, we first need to understand the general term of a binomial expansion. For any binomial expression of the form $(a+b)^n$, the general term, often denoted as $T_{r+1}$, is given by the formula:

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

where:

• $n$ is the power to which the binomial is raised.
• $r$ is the index of the term (starting from $r=0$ for the first term), and $r$ must be a non-negative integer ($0 \le r \le n$).
• ${ }^n C_r = \frac{n!}{r!(n-r)!}$ is the binomial coefficient.
• $a$ is the first term of the binomial.
• $b$ is the second term of the binomial.

In our given problem, the expression is $\left(3 x^{2}-\frac{1}{2 x^{5}}\right)^{7}$. Comparing this with $(a+b)^n$, we can identify:

• $a = 3x^2$
• $b = -\frac{1}{2x^5}$
• $n = 7$

2. Determining the General Term ($T_{r+1}$)

Now, we substitute these values into the general term formula:

$T_{r+1} = { }^7 C_r (3x^2)^{7-r} \left(-\frac{1}{2x^5}\right)^r$

To simplify this, we separate the constant parts from the $x$ parts:

$T_{r+1} = { }^7 C_r (3)^{7-r} (x^2)^{7-r} \left(-\frac{1}{2}\right)^r \left(\frac{1}{x^5}\right)^r$

Using the exponent rules $(x^p)^q = x^{pq}$ and $\frac{1}{x^p} = x^{-p}$:

$T_{r+1} = { }^7 C_r (3)^{7-r} x^{2(7-r)} \left(-\frac{1}{2}\right)^r x^{-5r}$

Combining the powers of $x$:

$T_{r+1} = { }^7 C_r (3)^{7-r} \left(-\frac{1}{2}\right)^r x^{14-2r-5r}$

$T_{r+1} = { }^7 C_r (3)^{7-r} \left(-\frac{1}{2}\right)^r x^{14-7r}$

This is the simplified general term of the expansion.

3. Finding the Term Independent of $x$ (Constant Term)

A term is considered "constant" or "independent of $x$" if it does not contain the variable $x$. This means that the power of $x$ in that term must be equal to zero.

From our general term $T_{r+1}$, the power of $x$ is $14-7r$. We set this exponent to zero to find the value of $r$ that corresponds to the constant term:

$14 - 7r = 0$

Now, we solve for $r$:

$14 = 7r$ $r = \frac{14}{7}$ $r = 2$

Since $r=2$ is a non-negative integer and $0 \le 2 \le 7$, this is a valid value for $r$.

Tip: Always ensure that the value of $r$ you find is a non-negative integer within the range of $0$ to $n$. If $r$ turns out to be a fraction or a negative number, it means there is no constant term in the expansion.

4. Calculating the Constant Term ($\alpha$)

Now that we have found $r=2$, we substitute this value back into the constant part of our simplified general term (the part without $x^{14-7r}$) to find the constant term, which is given as $\alpha$.

$\alpha = { }^7 C_2 (3)^{7-2} \left(-\frac{1}{2}\right)^2$

Let's calculate each part:

• Binomial Coefficient: ${ }^7 C_2 = \frac{7!}{2!(7-2)!} = \frac{7!}{2!5!} = \frac{7 \times 6}{2 \times 1} = 7 \times 3 = 21$
• First part's coefficient: $(3)^{7-2} = (3)^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$
• Second part's coefficient: $\left(-\frac{1}{2}\right)^2 = \left(-1 \times \frac{1}{2}\right)^2 = (-1)^2 \times \left(\frac{1}{2}\right)^2 = 1 \times \frac{1}{4} = \frac{1}{4}$

Common Mistake: Be careful with negative signs raised to powers. An even power of a negative number results in a positive number.

Now, multiply these values to find $\alpha$:

$\alpha = 21 \times 243 \times \frac{1}{4}$ $\alpha = 5103 \times \frac{1}{4}$ $\alpha = \frac{5103}{4}$ $\alpha = 1275.75$

5. Evaluating the Greatest Integer Function ($[\alpha]$)

The problem asks for the value of $[\alpha]$, where $[t]$ denotes the greatest integer less than or equal to $t$. This is also known as the floor function.

For $\alpha = 1275.75$, we need to find the greatest integer that is less than or equal to $1275.75$.

$[\alpha] = [1275.75]$ $[\alpha] = 1275$

Therefore, the value of $[\alpha]$ is $1275$.

6. Key Takeaways

This problem demonstrates a standard application of the Binomial Theorem. The key steps are:

1. Identify $a, b, n$ from the given binomial expansion.
2. Write the general term ($T_{r+1}$) and simplify it, carefully separating constant and variable parts.
3. Set the power of the variable ($x$) to zero to find the term independent of $x$ (constant term). Solve for $r$.
4. Substitute the value of $r$ back into the constant part of $T_{r+1}$ to calculate the desired term.
5. Apply any additional functions (like the greatest integer function) as required by the problem. Paying close attention to algebraic manipulations, especially with exponents and signs, is crucial for accuracy.