Let `alpha_(r )` where `r=1,2,3,4,.....,100` be the roots of `sum_(k=0)^(100)(z)^(k)=0` . If `sum_(r=1)^(100)(1)/((alpha_(r )-1))=10lambda` , then`|lambda|` is equal to _________.

Answer

To solve the problem, we need to find the roots of the polynomial given by the equation:

\[ \sum_{k=0}^{100} z^k = 0 \]

### Step 1: Identify the Polynomial
The sum of a geometric series can be expressed as:

\[ \frac{1 - z^{101}}{1 - z} = 0 \]

This implies that:

\[ 1 - z^{101} = 0 \quad \text{or} \quad 1 - z = 0 \]

The roots of the equation \(1 - z^{101} = 0\) are the 101st roots of unity, which can be expressed as:

\[ z = e^{2\pi i k / 101} \quad \text{for } k = 0, 1, 2, \ldots, 100 \]

### Step 2: Identify the Roots
The roots of the polynomial are:

\[ \alpha_r = e^{2\pi i r / 101} \quad \text{for } r = 1, 2, \ldots, 100 \]

### Step 3: Calculate the Required Sum
We need to evaluate the sum:

\[ \sum_{r=1}^{100} \frac{1}{\alpha_r - 1} \]

### Step 4: Substitute the Roots
Substituting \(\alpha_r\):

\[ \sum_{r=1}^{100} \frac{1}{e^{2\pi i r / 101} - 1} \]

### Step 5: Simplify the Expression
To simplify the expression, we can use the fact that:

\[ e^{2\pi i r / 101} - 1 = e^{\pi i r / 101} (e^{\pi i r / 101} - e^{-\pi i r / 101}) = e^{\pi i r / 101} \cdot 2i \sin\left(\frac{\pi r}{101}\right) \]

Thus, we can rewrite our sum as:

\[ \sum_{r=1}^{100} \frac{1}{e^{2\pi i r / 101} - 1} = \sum_{r=1}^{100} \frac{1}{e^{\pi i r / 101} \cdot 2i \sin\left(\frac{\pi r}{101}\right)} \]

### Step 6: Factor Out Constants
Factoring out the constant \(2i\):

\[ \frac{1}{2i} \sum_{r=1}^{100} \frac{1}{e^{\pi i r / 101} \sin\left(\frac{\pi r}{101}\right)} \]

### Step 7: Evaluate the Sum
The sum can be evaluated using properties of roots of unity and symmetry. The sum of the residues at the poles will yield:

\[ \sum_{r=1}^{100} \frac{1}{\alpha_r - 1} = \frac{101}{2} \]

### Step 8: Relate to Given Equation
From the problem, we know:

\[ \sum_{r=1}^{100} \frac{1}{\alpha_r - 1} = 10\lambda \]

Setting this equal to our derived result:

\[ 10\lambda = \frac{101}{2} \]

### Step 9: Solve for \(\lambda\)
Solving for \(\lambda\):

\[ \lambda = \frac{101}{20} \]

### Step 10: Find the Magnitude
The magnitude of \(\lambda\) is:

\[ |\lambda| = \frac{101}{20} \]

Thus, the final answer is:

\[ \boxed{5.05} \]