To find the 10th partial sum of the arithmetic sequence defined by \( a_n = \frac{1}{2}n - \frac{1}{2} \), we first need to identify the first term and the common difference.
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Calculate the first term \( a_1 \): \[ a_1 = \frac{1}{2}(1) - \frac{1}{2} = \frac{1}{2} - \frac{1}{2} = 0 \]
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Calculate the second term \( a_2 \): \[ a_2 = \frac{1}{2}(2) - \frac{1}{2} = 1 - \frac{1}{2} = \frac{1}{2} \]
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Calculate the third term \( a_3 \): \[ a_3 = \frac{1}{2}(3) - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1 \]
From the calculations, we can see that the sequence starts from \( a_1 = 0 \) and goes \( 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots \).
Now we observe that this is indeed an arithmetic sequence where:
- The first term \( a_1 = 0 \)
- The common difference: \[ a_2 - a_1 = \frac{1}{2} - 0 = \frac{1}{2} \]
The \( n \)-th term of the arithmetic sequence can also be expressed as: \[ a_n = a_1 + (n-1)d = 0 + (n-1)\cdot\frac{1}{2} = \frac{1}{2}(n-1) \]
Now we'll calculate the 10th term \( a_{10} \): \[ a_{10} = \frac{1}{2}(10-1) = \frac{1}{2}(9) = \frac{9}{2} = 4.5 \]
To find the 10th partial sum \( S_{10} \), we use the formula for the sum of the first \( n \) terms of an arithmetic sequence: \[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]
In this case:
- \( n = 10 \)
- \( a_1 = 0 \)
- \( a_{10} = 4.5 \)
So, \[ S_{10} = \frac{10}{2} \cdot (0 + 4.5) = 5 \cdot 4.5 = 22.5 \]
Therefore, the 10th partial sum is:
\[ \boxed{22.5} \]