Sure, let's break down the problem step by step.
We need to calculate the sum of the following series:
[tex]\[ 55 \left( \frac{1}{11} \right) + 55 \left( \frac{2}{11} \right) + 55 \left( \frac{3}{11} \right) + \ldots + 55 \left( \frac{10}{11} \right). \][/tex]
First, let's understand each term in this sequence:
[tex]\[ 55 \left( \frac{1}{11} \right), 55 \left( \frac{2}{11} \right), 55 \left( \frac{3}{11} \right), \ldots, 55 \left( \frac{10}{11} \right). \][/tex]
Let's break these fractions down individually, where each term is:
[tex]\[ \frac{55 \cdot 1}{11}, \frac{55 \cdot 2}{11}, \frac{55 \cdot 3}{11}, \ldots, \frac{55 \cdot 10}{11}. \][/tex]
Simplifying each term:
[tex]\[ 55 \left( \frac{1}{11} \right) = 5. \][/tex]
[tex]\[ 55 \left( \frac{2}{11} \right) = 10. \][/tex]
[tex]\[ 55 \left( \frac{3}{11} \right) = 15. \][/tex]
[tex]\[ 55 \left( \frac{4}{11} \right) = 20. \][/tex]
[tex]\[ 55 \left( \frac{5}{11} \right) = 25. \][/tex]
[tex]\[ 55 \left( \frac{6}{11} \right) = 30. \][/tex]
[tex]\[ 55 \left( \frac{7}{11} \right) = 35. \][/tex]
[tex]\[ 55 \left( \frac{8}{11} \right) = 40. \][/tex]
[tex]\[ 55 \left( \frac{9}{11} \right) = 45. \][/tex]
[tex]\[ 55 \left( \frac{10}{11} \right) = 50. \][/tex]
Next, we sum all these terms:
[tex]\[ 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50. \][/tex]
Grouping them to make the calculation easier:
[tex]\[ (5 + 50) + (10 + 45) + (15 + 40) + (20 + 35) + (25 + 30). \][/tex]
Each of these pairs sums up to 55:
[tex]\[ 55 + 55 + 55 + 55 + 55 = 5 \times 55 = 275. \][/tex]
Therefore, the sum of the given series is:
[tex]\[ \boxed{275} \][/tex]
