the binomial expansion of (1+kx)^n is given by 1+(9x)/2+15k^2x^2+...+k^nx^n. Find n and k

To find the values of \( n \) and \( k \) from the given binomial expansion of \((1+kx)^n\), we can compare the coefficients of the expansion with those provided in the problem.

The binomial expansion of \((1+kx)^n\) is given by:

\[
(1 + kx)^n = 1 + \binom{n}{1}kx + \binom{n}{2}k^2x^2 + \cdots + \binom{n}{n}k^n x^n
\]

Comparing this with the given expansion \( 1 + \frac{9x}{2} + 15k^2x^2 + \cdots + k^n x^n \):

1. **For the coefficient of \(x\):**

   \[
   \binom{n}{1}k = \frac{9}{2}
   \]

   Simplifying, we have:
   \[
   nk = \frac{9}{2} \quad \text{(since \( \binom{n}{1} = n \))}
   \]

2. **For the coefficient of \(x^2\):**

   \[
   \binom{n}{2}k^2 = 15k^2
   \]

   Simplifying, we have:
   \[
   \frac{n(n-1)}{2} = 15
   \]
   \[
   n(n-1) = 30
   \]

   This is a quadratic equation: 
   \[
   n^2 - n - 30 = 0
   \]

   Solving this quadratic equation using the quadratic formula:
   \[
   n = \frac{-(-1) \pm \sqrt{(-1)^2 + 4(1)(30)}}{2(1)}
   \]
   \[
   n = \frac{1 \pm \sqrt{1 + 120}}{2}
   \]
   \[
   n = \frac{1 \pm \sqrt{121}}{2}
   \]
   \[
   n = \frac{1 \pm 11}{2}
   \]

   So, \( n = 6 \) or \( n = -5 \). Since \( n \) must be a positive integer, we have \( n = 6 \).

3. **Substituting \( n = 6 \) back into the equation \( nk = \frac{9}{2} \):**

   \[
   6k = \frac{9}{2}
   \]
   \[
   k = \frac{9}{12} = \frac{3}{4}
   \]

Thus, the values of \( n \) and \( k \) are:

\[
n = 6, \quad k = \frac{3}{4}
\]

Would you like further details on the binomial expansion, or do you have any other questions?

Here are 5 related questions:

1. How do you derive the general term in a binomial expansion?
2. What is the significance of the binomial coefficient \(\binom{n}{r}\)?
3. How can the binomial theorem be applied to approximate functions?
4. What happens to the binomial expansion when \( k = 1 \)?
5. Can you use the binomial expansion to calculate probabilities in a binomial distribution?

**Tip:** When solving quadratic equations, always check the discriminant (\(\Delta = b^2 - 4ac\)) first to determine the nature of the roots.

The binomial expansion of (1+kx)^n is given by 1+(9x)/2+15k^2x^2+...+k^nx^n. Find n and k.

Explore how to determine the values of n and k from the given binomial expansion (1+kx)^n. This solution covers the application of the binomial theorem and solving quadratic equations, making it ideal for students in Grades 11-12.

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