Question

If in the expansion of \( (1 + x^2)^2(1 + x)^n \), the coefficients of \( x \), \( x^2 \), and \( x^3 \) are in arithmetic progression, then the sum of all possible values of \( n \) (where \( n \geq 3 \)) is:

Hint:

When dealing with binomial expansions, always remember to carefully track the powers of \( x \) and use the properties of arithmetic progressions to form relationships between the coefficients.

Answer 1

Correct Answer: 7

Step 1: Analyze the Expansion.
The given expression is \( (1 + x^2)^2(1 + x)^n \). First, expand \( (1 + x^2)^2 \) and then multiply it with \( (1 + x)^n \). \[ (1 + x^2)^2 = 1 + 2x^2 + x^4 \] Now, multiply this by \( (1 + x)^n \): \[ (1 + x^2)^2 (1 + x)^n = (1 + 2x^2 + x^4)(1 + x)^n \]
Step 2: Identify Coefficients of \( x \), \( x^2 \), and \( x^3 \).
Now, we need to find the coefficients of \( x \), \( x^2 \), and \( x^3 \) in the expansion of the product. To do this, apply the binomial theorem to expand \( (1 + x)^n \). - The coefficient of \( x \) is given by the term involving \( x \) from both expansions. - Similarly, find the coefficients of \( x^2 \) and \( x^3 \).
Step 3: Set up the Arithmetic Progression.
We are given that the coefficients of \( x \), \( x^2 \), and \( x^3 \) form an arithmetic progression. Let the coefficients be denoted as \( A_1 \), \( A_2 \), and \( A_3 \), respectively. Since they form an arithmetic progression, we have the condition: \[ 2A_2 = A_1 + A_3 \] Using the expressions for the coefficients of \( x \), \( x^2 \), and \( x^3 \), solve for \( n \).
Step 4: Solve for the Possible Values of \( n \).
After solving the equations and simplifying, we find that the sum of all possible values of \( n \) is 7. Final Answer: \[ \boxed{7} \]