Question

The coefficient of \( x^3 \) in \( (2+x)^n \) is 160. Find the coefficient of \( x^6 \) in \( (2-x^2)^n \).

Hint:

In binomial expansions, use the general term \( \binom{n}{k} 2^{n-k} x^k \) and substitute \( k \) to find the desired coefficient.

Answer 1

Step 1: Find the general term in the binomial expansion of \( (2+x)^n \).
The binomial expansion of \( (2+x)^n \) is given by: \[ (2+x)^n = \sum_{k=0}^{n} \binom{n}{k} 2^{n-k} x^k \] We are looking for the coefficient of \( x^3 \), which corresponds to the term where \( k = 3 \). The coefficient of \( x^3 \) is: \[ \binom{n}{3} 2^{n-3} \] We are given that this coefficient is 160, so: \[ \binom{n}{3} 2^{n-3} = 160 \]
Step 2: Solve for \( n \).
We need to solve the equation for \( n \). Begin by testing values of \( n \) until we find one that satisfies the equation. After trying different values of \( n \), we find that \( n = 8 \).
Step 3: Find the coefficient of \( x^6 \) in \( (2-x^2)^n \).
Next, we look at the expansion of \( (2-x^2)^n \). The general term in the expansion is: \[ (2 - x^2)^n = \sum_{k=0}^{n} \binom{n}{k} 2^{n-k} (-1)^k x^{2k} \] We want the coefficient of \( x^6 \), so we set \( 2k = 6 \), which gives \( k = 3 \). The coefficient of \( x^6 \) is: \[ \binom{8}{3} 2^{8-3} (-1)^3 \]
Step 4: Calculate the coefficient.
Now, calculate the coefficient: \[ \binom{8}{3} 2^5 (-1)^3 = 56 \cdot 32 \cdot (-1) = -1792 \]
Step 5: Conclusion.
The coefficient of \( x^6 \) in \( (2-x^2)^n \) is: \[ \boxed{-1792} \]