The coefficient of `x^7` in
`(1 + 2x + 3x^2 + 4x^3 + …… "to " oo)`

Options

A. 0
B. 6
C. 8
D. 36

Answer

To find the coefficient of \( x^7 \) in the series \( 1 + 2x + 3x^2 + 4x^3 + \ldots \), we can follow these steps:

### Step 1: Identify the General Term
The series can be expressed in terms of its general term. The \( n \)-th term of the series can be represented as:
\[ T_n = n \cdot x^{n-1} \]
where \( n \) starts from 1.

### Step 2: Determine the Value of \( n \)
We need to find the coefficient of \( x^7 \). In our general term \( T_n = n \cdot x^{n-1} \), we want the exponent of \( x \) to be 7. This means we need:
\[ n - 1 = 7 \]
Solving for \( n \):
\[ n = 8 \]

### Step 3: Calculate the Coefficient
Now, we substitute \( n = 8 \) into the general term to find the coefficient:
\[ T_8 = 8 \cdot x^{8-1} = 8 \cdot x^7 \]
The coefficient of \( x^7 \) is therefore 8.

### Final Answer
Thus, the coefficient of \( x^7 \) in the series is:
\[ \boxed{8} \]