If the coefficient of x to the power 4 and x to the power 5 in the expansion of (3+kx)to the power 10 are equal find the value of k.

Let's solve the problem step by step.

The given expression is \((3 + kx)^{10}\).

We need to find the coefficient of \(x^4\) and \(x^5\) in the expansion of this expression and set them equal to find the value of \(k\).

### Step 1: Find the general term in the expansion
The general term in the expansion of \((3 + kx)^{10}\) using the Binomial Theorem is given by:

\[
T_{r+1} = \binom{10}{r} \cdot 3^{10-r} \cdot (kx)^r
\]

Simplifying this, we get:

\[
T_{r+1} = \binom{10}{r} \cdot 3^{10-r} \cdot k^r \cdot x^r
\]

### Step 2: Identify the coefficients of \(x^4\) and \(x^5\)
- **For \(x^4\):** The coefficient is obtained by setting \(r = 4\):

\[
\text{Coefficient of } x^4 = \binom{10}{4} \cdot 3^{6} \cdot k^4
\]

- **For \(x^5\):** The coefficient is obtained by setting \(r = 5\):

\[
\text{Coefficient of } x^5 = \binom{10}{5} \cdot 3^{5} \cdot k^5
\]

### Step 3: Set the coefficients equal to each other
Since the coefficients are equal, we have:

\[
\binom{10}{4} \cdot 3^{6} \cdot k^4 = \binom{10}{5} \cdot 3^{5} \cdot k^5
\]

### Step 4: Simplify the equation
Divide both sides by \(3^{5} \cdot k^4\):

\[
\binom{10}{4} \cdot 3 = \binom{10}{5} \cdot k
\]

Now, calculate the binomial coefficients:

\[
\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

\[
\binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
\]

So the equation becomes:

\[
210 \cdot 3 = 252 \cdot k
\]

\[
630 = 252k
\]

### Step 5: Solve for \(k\)

\[
k = \frac{630}{252} = \frac{105}{42} = \frac{15}{6} = \frac{5}{2}
\]

So, the value of \(k\) is \( \boxed{\frac{5}{2}} \).

Would you like further details or have any questions?

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**Related Questions:**
1. How do we calculate binomial coefficients?
2. What happens if the coefficients of \(x^3\) and \(x^5\) are equal?
3. How does the Binomial Theorem apply to negative or fractional exponents?
4. Can the value of \(k\) be negative in this problem? Why or why not?
5. What would happen if we set the coefficients of \(x^4\) and \(x^6\) equal?

**Tip:** Always simplify equations as much as possible before solving for variables; it reduces complexity and potential errors.

Discover how to find the value of k in the expansion of (3+kx)^10 using the Binomial Theorem. Step-by-step solution for calculating coefficients of x^4 and x^5, setting them equal, and solving for k. Suitable for advanced high school students.

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