Let $(2 - x)^9 = a_0 + a_1x + a_2x^2 + \dots + a_9x^9$. Then the value of $a_1 + a_2 + a_3 + \dots + a_8$ is equal to

Question

If $ A $ and $ B $ are the two real values of $ k $ for which the system of equations $ x + 2y + z = 1 $, $ x + 3y + 4z = k $, $ x + 5y + 10z = k^2 $ is consistent, then $ A + B = $: (a) 3

Answer 1

Step 1: Understanding the Concept:
The sum of coefficients in a polynomial expansion can be found by substituting specific values for $x$.
We are asked for the sum of the coefficients excluding the first ($a_0$) and last ($a_9$) terms.
Step 2: Key Formula or Approach:
Set $x = 1$ to find the total sum of all coefficients.
Explicitly calculate $a_0$ and $a_9$, and subtract them from the total sum.
Step 3: Detailed Explanation:
Substitute $x = 1$ into the equation:
\[ (2 - 1)^9 = a_0 + a_1(1) + a_2(1)^2 + \dots + a_9(1)^9 \] \[ 1^9 = a_0 + a_1 + a_2 + \dots + a_9 = 1 \] Now, calculate $a_0$, which is the constant term (when $x = 0$):
\[ a_0 = (2)^9 = 512 \] Calculate $a_9$, the coefficient of the highest power term ($x^9$):
The $x^9$ term is $(-x)^9 = -x^9$, so $a_9 = -1$.
We need the sum $S = a_1 + a_2 + \dots + a_8$.
We know:
\[ a_0 + S + a_9 = 1 \] \[ 512 + S - 1 = 1 \] \[ S + 511 = 1 \] \[ S = 1 - 511 = -510 \] Step 4: Final Answer:
The sum is -510.

Let $(2 - x)^9 = a_0 + a_1x + a_2x^2 + \dots + a_9x^9$. Then the value of $a_1 + a_2 + a_3 + \dots + a_8$ is equal to

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