If `1+2sin x+3sin^(2)x+4sin^(3)x+...` upto infinite terms = 4 and number of solutions of the equation in `[ (-3pi)/(2), 4pi] ` is k.
Sum of all internal angles of a k-sided regular polygon is

Options

A. `5pi`
B. `4pi`
C. `3pi`
D. `2pi`

Answer

To solve the problem step by step, we start with the given infinite series:

### Step 1: Set Up the Equation
The series is given as:
\[ 1 + 2\sin x + 3\sin^2 x + 4\sin^3 x + \ldots = 4 \]
This can be expressed as:
\[ \sum_{n=1}^{\infty} n \sin^{n-1} x = 4 \]

### Step 2: Use the Formula for the Sum of Infinite Series
The sum of the series can be derived using the formula for the sum of an infinite geometric series. We know that:
\[ \sum_{n=0}^{\infty} n r^n = \frac{r}{(1 - r)^2} \quad \text{for } |r| < 1 \]
In our case, we can rewrite the series:
\[ \sum_{n=1}^{\infty} n \sin^{n} x = \sin x \sum_{n=0}^{\infty} n \sin^{n-1} x = \frac{\sin x}{(1 - \sin x)^2} \]
Thus, we have:
\[ \frac{\sin x}{(1 - \sin x)^2} = 4 \]

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ \sin x = 4(1 - \sin x)^2 \]
Expanding the right-hand side:
\[ \sin x = 4(1 - 2\sin x + \sin^2 x) \]
This simplifies to:
\[ \sin x = 4 - 8\sin x + 4\sin^2 x \]
Rearranging gives:
\[ 4\sin^2 x - 9\sin x + 4 = 0 \]

### Step 4: Solving the Quadratic Equation
Using the quadratic formula \( \sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4, b = -9, c = 4 \):
\[ \sin x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \]
Calculating the discriminant:
\[ \sqrt{81 - 64} = \sqrt{17} \]
Thus, we have:
\[ \sin x = \frac{9 \pm \sqrt{17}}{8} \]

### Step 5: Finding Valid Solutions
Now we need to check the values of \( \sin x \):
1. \( \sin x = \frac{9 + \sqrt{17}}{8} \) (Check if this is ≤ 1)
2. \( \sin x = \frac{9 - \sqrt{17}}{8} \) (Check if this is ≥ -1)

Calculating:
- For \( \frac{9 + \sqrt{17}}{8} \), since \( \sqrt{17} \approx 4.123 \), we have:
\[ \frac{9 + 4.123}{8} \approx \frac{13.123}{8} \approx 1.640 \quad (\text{not valid}) \]
- For \( \frac{9 - \sqrt{17}}{8} \):
\[ \frac{9 - 4.123}{8} \approx \frac{4.877}{8} \approx 0.609625 \quad (\text{valid}) \]

### Step 6: Finding the Number of Solutions in the Interval
The valid solution is \( \sin x = \frac{9 - \sqrt{17}}{8} \). We need to find the number of solutions in the interval \( \left[-\frac{3\pi}{2}, 4\pi\right] \).

The general solutions for \( \sin x = k \) are:
\[ x = n\pi + (-1)^n \arcsin(k) \]
Calculating the number of solutions in the given interval:
1. The period of sine is \( 2\pi \).
2. Find the range of \( n \) such that \( -\frac{3\pi}{2} \leq n\pi + (-1)^n \arcsin(k) \leq 4\pi \).

After calculating, we find that there are **5 solutions** in the interval.

### Step 7: Calculate the Sum of Internal Angles of a k-sided Polygon
The sum of the internal angles of a k-sided polygon is given by:
\[ \text{Sum} = (k - 2) \times 180^\circ \]
For \( k = 5 \):
\[ \text{Sum} = (5 - 2) \times 180 = 3 \times 180 = 540^\circ \]

### Final Answer
The sum of all internal angles of a 5-sided regular polygon is \( 540^\circ \).