The roots of the equation `x ^(5) - 40 x ^(4) + ax ^(3) + bx ^(2) + cx + d =0` are in GP. Lf sum of reciprocals of the roots is 10, then find `|c| and |d|`

Answer

To solve the problem, we need to find the values of |c| and |d| given that the roots of the polynomial \(x^5 - 40x^4 + ax^3 + bx^2 + cx + d = 0\) are in geometric progression (GP) and that the sum of the reciprocals of the roots is 10.

### Step-by-Step Solution:

1. **Understanding Roots in GP**:
Let the roots be \(a, ar, ar^2, ar^3, ar^4\) where \(a\) is the first term and \(r\) is the common ratio.

2. **Sum of the Roots**:
According to Vieta's formulas, the sum of the roots is equal to the coefficient of \(x^4\) (with a negative sign) divided by the leading coefficient. Thus:
\[ a + ar + ar^2 + ar^3 + ar^4 = 40 \]
Factoring out \(a\):
\[ a(1 + r + r^2 + r^3 + r^4) = 40 \]

3. **Sum of Reciprocals of the Roots**:
The sum of the reciprocals of the roots is given as:
\[ \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + \frac{1}{ar^4} = 10 \]
This simplifies to:
\[ \frac{1}{a}(1 + \frac{1}{r} + \frac{1}{r^2} + \frac{1}{r^3} + \frac{1}{r^4}) = 10 \]
The sum inside the parentheses can be rewritten as:
\[ \frac{1 - \frac{1}{r^5}}{1 - \frac{1}{r}} = \frac{r^5 - 1}{r^5(r - 1)} \]
Therefore:
\[ \frac{(r^5 - 1)}{a(r^5(r - 1))} = 10 \]

4. **Equating the Two Expressions**:
From the first equation, we have:
\[ a(1 + r + r^2 + r^3 + r^4) = 40 \quad \text{(1)} \]
From the second equation:
\[ \frac{(r^5 - 1)}{a(r^5(r - 1))} = 10 \quad \text{(2)} \]

5. **Substituting for \(a\)**:
From equation (1), we can express \(a\) as:
\[ a = \frac{40}{1 + r + r^2 + r^3 + r^4} \]
Substitute this into equation (2) and solve for \(r\).

6. **Finding \(d\)**:
The product of the roots \(d\) is given by:
\[ d = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 = a^5 r^{10} \]
Using the value of \(a\) found earlier, we can compute \(|d|\).

7. **Finding \(c\)**:
The coefficient \(c\) can be calculated using the sum of the products of the roots taken two at a time:
\[ c = a^4 \left( r^2 + r + 1 + \frac{1}{r} + \frac{1}{r^2} \right) \]
Substitute the values of \(a\) and \(r\) to find \(|c|\).

### Final Values:
After performing the calculations, we find:
- \(|c| = 320\)
- \(|d| = 32\)