The value of
`152 ( sin 30^@ + 2cos^2 45^@ + 3 sin 30^@ + 4 cos^2 45^@ + …+ 17 sin30^@ + 18 cos^2 45^@)` is

Options

A. an integer but not a perfect square
B. a rational number but not an Integer
C. a perfect square of an integer
D. irrational

Answer

To solve the given expression \( 152 \left( \sin 30^\circ + 2 \cos^2 45^\circ + 3 \sin 30^\circ + 4 \cos^2 45^\circ + \ldots + 17 \sin 30^\circ + 18 \cos^2 45^\circ \right) \), we will break it down step by step.

### Step 1: Identify the values of trigonometric functions
We know:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \) so \( \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)

### Step 2: Rewrite the expression
Now substituting these values into the expression:
\[ = 152 \left( \frac{1}{2} + 2 \cdot \frac{1}{2} + 3 \cdot \frac{1}{2} + 4 \cdot \frac{1}{2} + \ldots + 17 \cdot \frac{1}{2} + 18 \cdot \frac{1}{2} \right) \]

### Step 3: Factor out common terms
Factoring out \( \frac{1}{2} \):
\[ = 152 \left( \frac{1}{2} \left( 1 + 2 + 3 + 4 + \ldots + 17 + 18 \right) \right) \]

### Step 4: Calculate the sum of the series
The sum of the first \( n \) natural numbers is given by the formula:
\[ S_n = \frac{n(n + 1)}{2} \]
For \( n = 18 \):
\[ S_{18} = \frac{18 \times 19}{2} = 171 \]

### Step 5: Substitute back into the expression
Now substituting \( S_{18} \) back into the expression:
\[ = 152 \left( \frac{1}{2} \times 171 \right) \]
\[ = 152 \times \frac{171}{2} \]
\[ = 76 \times 171 \]

### Step 6: Calculate \( 76 \times 171 \)
Calculating \( 76 \times 171 \):
\[ 76 \times 171 = 12996 \]

### Final Result
Thus, the value of the expression is:
\[ \boxed{12996} \]