Question

If $A$ denotes the sum of all the coefficients in the expansion of $(1 - 3x + 10x^2)^n$ and $B$ denotes the sum of all the coefficients in the expansion of $(1 + x^2)^n$, then:

• A. $A = B^3$
• B. $3A = B$
• C. $B = A^3$
• D. $A = 3B$

Answer 1

Answer: (A)

$A = B^3$

Answer 2

To find the sums $A$ and $B$, we calculate the sum of all coefficients by setting $x = 1$ in each expansion.

Step 1. Calculate $A$

Substitute $x = 1$ in $(1 - 3x + 10x^2)^n$:$A = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n = (1 - 3 + 10)^n = 8^n$Therefore, $A = 8^n$.

Step 2. Calculate $B$

Substitute $x = 1$ in $(1 + x^2)^n$:$B = (1 + 1^2)^n = 2^n$Thus, $B = 2^n$.

Step 3. Find the Relationship Between $A$ and $B$

Since $A = 8^n$ and $B = 2^n$, we can write:$A = (2^n)^3 = B^3$Therefore, $A = B^3$.