The coefficient of x^3 is four times larger than the coefficient of x^2 in the expansion of (2+x)^n. Find n

To solve the problem, we need to determine the value of \( n \) for which the coefficient of \( x^3 \) is four times the coefficient of \( x^2 \) in the expansion of \( (2 + x)^n \).

### Step 1: Use the Binomial Theorem
The binomial expansion of \( (2 + x)^n \) is given by:

\[
(2 + x)^n = \sum_{k=0}^{n} \binom{n}{k} 2^{n-k} x^k
\]

Here, the general term in the expansion is:

\[
T_k = \binom{n}{k} 2^{n-k} x^k
\]

### Step 2: Identify the coefficients of \( x^3 \) and \( x^2 \)
- The coefficient of \( x^2 \) is given by:

\[
\text{Coefficient of } x^2 = \binom{n}{2} 2^{n-2}
\]

- The coefficient of \( x^3 \) is given by:

\[
\text{Coefficient of } x^3 = \binom{n}{3} 2^{n-3}
\]

### Step 3: Set up the equation
According to the problem, the coefficient of \( x^3 \) is four times the coefficient of \( x^2 \). Therefore, we set up the equation:

\[
\binom{n}{3} 2^{n-3} = 4 \times \binom{n}{2} 2^{n-2}
\]

### Step 4: Simplify the equation
First, divide both sides by \( 2^{n-3} \):

\[
\binom{n}{3} = 4 \times \binom{n}{2} \times 2
\]

This simplifies to:

\[
\binom{n}{3} = 8 \times \binom{n}{2}
\]

### Step 5: Use the binomial coefficients
Recall that:

\[
\binom{n}{3} = \frac{n(n-1)(n-2)}{6}, \quad \binom{n}{2} = \frac{n(n-1)}{2}
\]

Substitute these into the equation:

\[
\frac{n(n-1)(n-2)}{6} = 8 \times \frac{n(n-1)}{2}
\]

### Step 6: Solve for \( n \)
Cancel \( n(n-1) \) from both sides (assuming \( n \neq 0 \) and \( n \neq 1 \)):

\[
\frac{n-2}{6} = 8
\]

Multiply both sides by 6:

\[
n - 2 = 48
\]

Finally, solve for \( n \):

\[
n = 50
\]

### Answer:
The value of \( n \) is \( \boxed{50} \).

Would you like more details, or do you have any further questions?

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Here are 5 related questions to further explore this topic:
1. How does the binomial coefficient formula work, and why is it important in expansions?
2. What is the general pattern of coefficients in binomial expansions?
3. How would the problem change if the expression was \( (1+x)^n \) instead of \( (2+x)^n \)?
4. What are other methods to find the coefficients in binomial expansions?
5. Can you apply this method to higher powers, such as finding the coefficient ratio between \( x^4 \) and \( x^3 \)?

**Tip:** When dealing with binomial expansions, always check the relationships between coefficients, as they can provide valuable clues to solve the problem efficiently.

Learn how to find the value of n in the expansion of (2+x)^n where the coefficient of x^3 is four times the coefficient of x^2. Detailed step-by-step solution using binomial coefficients and theorem. Suitable for Grades 11-12 students.

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