If `D_r=|{:(r,15,8),(r^2,35,9),(r^3,25,10):}|`, then the value of `root(5)(((-1/100)sum_(r=1)^5D_r)-37)` is equal to

Options

A. 5
B. 2
C. 9
D. 3

Answer

To solve the problem, we need to evaluate the expression given in the question step by step. Let's break it down:

### Step 1: Define the Determinant \( D_r \)
The determinant \( D_r \) is given by:
\[ D_r = \begin{vmatrix} r & 15 & 8 \\ r^2 & 35 & 9 \\ r^3 & 25 & 10 \end{vmatrix} \]

### Step 2: Calculate the Determinant \( D_r \)
Using the formula for the determinant of a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]
we can substitute \( a = r, b = 15, c = 8, d = r^2, e = 35, f = 9, g = r^3, h = 25, i = 10 \).

Calculating the determinant:
\[ D_r = r(35 \cdot 10 - 9 \cdot 25) - 15(r^2 \cdot 10 - 9 \cdot r^3) + 8(r^2 \cdot 25 - 35 \cdot r^3) \]

Calculating each term:
1. \( 35 \cdot 10 - 9 \cdot 25 = 350 - 225 = 125 \)
2. \( r(125) = 125r \)

3. \( r^2 \cdot 10 - 9 \cdot r^3 = 10r^2 - 9r^3 \)
4. \( -15(10r^2 - 9r^3) = -150r^2 + 135r^3 \)

5. \( r^2 \cdot 25 - 35 \cdot r^3 = 25r^2 - 35r^3 \)
6. \( 8(25r^2 - 35r^3) = 200r^2 - 280r^3 \)

Combining all terms:
\[ D_r = 125r + (-150r^2 + 135r^3) + (200r^2 - 280r^3) \]
\[ D_r = 125r + (-150r^2 + 200r^2) + (135r^3 - 280r^3) \]
\[ D_r = 125r + 50r^2 - 145r^3 \]

### Step 3: Calculate the Summation \( \sum_{r=1}^{5} D_r \)
Now we need to compute:
\[ \sum_{r=1}^{5} D_r = \sum_{r=1}^{5} (125r + 50r^2 - 145r^3) \]
This can be separated into three summations:
\[ = 125 \sum_{r=1}^{5} r + 50 \sum_{r=1}^{5} r^2 - 145 \sum_{r=1}^{5} r^3 \]

Using the formulas:
- \( \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \)
- \( \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} \)
- \( \sum_{r=1}^{n} r^3 = \left( \frac{n(n+1)}{2} \right)^2 \)

For \( n = 5 \):
1. \( \sum_{r=1}^{5} r = \frac{5(5+1)}{2} = 15 \)
2. \( \sum_{r=1}^{5} r^2 = \frac{5(5+1)(2 \cdot 5 + 1)}{6} = 55 \)
3. \( \sum_{r=1}^{5} r^3 = \left( \frac{5(5+1)}{2} \right)^2 = 225 \)

Substituting these values:
\[ = 125(15) + 50(55) - 145(225) \]
Calculating each term:
1. \( 125 \cdot 15 = 1875 \)
2. \( 50 \cdot 55 = 2750 \)
3. \( 145 \cdot 225 = 32625 \)

Now, summing these:
\[ \sum_{r=1}^{5} D_r = 1875 + 2750 - 32625 = -27800 \]

### Step 4: Calculate the Final Expression
Now we need to evaluate:
\[ \sqrt[5]{\left(-\frac{1}{100} \sum_{r=1}^{5} D_r\right) - 37} \]
Substituting \( \sum_{r=1}^{5} D_r = -27800 \):
\[ = \sqrt[5]{-\frac{1}{100}(-27800) - 37} \]
\[ = \sqrt[5]{\frac{27800}{100} - 37} \]
\[ = \sqrt[5]{278 - 37} \]
\[ = \sqrt[5]{241} \]

### Step 5: Final Calculation
Since \( 241 \) does not have a perfect fifth root, we can approximate or conclude that the answer is simply \( \sqrt[5]{241} \).