If sum of series `(x+ka)+(x^2+(k-2)a)+(x^3+(k-4)a)+...9` terms is `((x^10-x-45a(x-1))/(x-1))` then value of `k` is:

Answer

To solve the problem, we need to find the value of \( k \) given the sum of the series:

\[ (x + ka) + (x^2 + (k-2)a) + (x^3 + (k-4)a) + \ldots \text{ (up to 9 terms)} \]

The sum of this series is given as:

\[ \frac{x^{10} - x - 45a(x-1)}{x-1} \]

### Step 1: Write down the series terms
The series consists of 9 terms. The first part of each term is \( x^n \) where \( n \) ranges from 1 to 9. The second part of each term is \( (k - (n-1) \cdot 2)a \).

So, we can express the series as:

\[ \sum_{n=1}^{9} \left( x^n + (k - (n-1) \cdot 2)a \right) \]

### Step 2: Separate the sums
We can separate the sums into two parts:

1. The sum of the first parts:
\[ S_1 = x + x^2 + x^3 + \ldots + x^9 \]
This is a geometric series.

2. The sum of the second parts:
\[ S_2 = \sum_{n=1}^{9} (k - (n-1) \cdot 2)a \]

### Step 3: Calculate \( S_1 \)
The sum \( S_1 \) can be calculated using the formula for the sum of a geometric series:

\[ S_1 = x \frac{x^9 - 1}{x - 1} \]

### Step 4: Calculate \( S_2 \)
For \( S_2 \):

\[ S_2 = \sum_{n=1}^{9} (k - (n-1) \cdot 2)a = 9ka - 2a \sum_{n=0}^{8} n \]

The sum \( \sum_{n=0}^{8} n = \frac{8 \cdot 9}{2} = 36 \).

Thus,

\[ S_2 = 9ka - 72a \]

### Step 5: Combine \( S_1 \) and \( S_2 \)
Now, combining both sums, we have:

\[ S = S_1 + S_2 = x \frac{x^9 - 1}{x - 1} + (9k - 72)a \]

### Step 6: Set the equation equal to the given sum
We set this equal to the given sum:

\[ x \frac{x^9 - 1}{x - 1} + (9k - 72)a = \frac{x^{10} - x - 45a(x-1)}{x-1} \]

### Step 7: Simplify the right-hand side
The right-hand side can be simplified as follows:

\[ \frac{x^{10} - x - 45a(x-1)}{x-1} = \frac{x^{10} - x + 45a - 45ax}{x-1} \]

### Step 8: Compare coefficients
Now, we compare the coefficients of \( x \) and \( a \) from both sides.

From the \( x \) terms:

\[ \frac{x^{10} - x}{x-1} \text{ gives } x^9 \text{ as the leading term.} \]

From the \( a \) terms:

\[ 9k - 72 = -45 \]

### Step 9: Solve for \( k \)
Now, we solve for \( k \):

\[ 9k - 72 = -45 \implies 9k = 27 \implies k = 3 \]

Thus, the value of \( k \) is:

\[ \boxed{3} \]