$$ ( 1 - \frac { 1 } { 2 } ) + ( 2 - \frac { 1 } { 2 } ) + \ldots + ( 10 - \frac { 1 } { 2 } ) = ? $$

Question

UpStudy AI · Accepted Answer

To evaluate the expression:

$$
(1 - \frac{1}{2}) + (2 - \frac{1}{2}) + \ldots + (10 - \frac{1}{2})
$$

We can break it down into two separate sums:

1. **Sum of integers from 1 to 10:**

$$
1 + 2 + 3 + \ldots + 10
$$

This is an arithmetic series with the sum formula:

$$
\text{Sum} = \frac{n(n + 1)}{2}
$$

For $$ n = 10 $$:

$$
\text{Sum} = \frac{10 \times 11}{2} = 55
$$

2. **Sum of $$-\frac{1}{2}$$ repeated 10 times:**

$$
- \frac{1}{2} - \frac{1}{2} - \ldots - \frac{1}{2} = 10 \times \left(-\frac{1}{2}\right) = -5
$$

**Combine both sums:**

$$
55 - 5 = 50
$$

**Final Answer:**

50
The final answer is 50.

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