13 Сен в 19:40
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To find the sum of the series from (-53) to (55), we can first observe that the numbers in this series form an arithmetic sequence.

The series can be expressed as:

[
S = -53 + (-52) + (-51) + \ldots + 0 + 1 + 2 + \ldots + 53 + 54 + 55
]

Step 1: Count the Numbers

The series includes all integers from (-53) to (55).

To find how many terms are in this series, we can calculate:

The smallest number is (-53).The largest number is (55).

The total count of integers from (-53) to (55) can be calculated by:

[
\text{Total count} = 55 - (-53) + 1 = 55 + 53 + 1 = 109
]

Step 2: Pair the Terms

Now we can pair the terms in the series to simplify the addition. We can pair terms as follows:

((-53) + 55 = 2)((-52) + 54 = 2)((-51) + 53 = 2)Continuing in this way up to ((0)), which remains unpaired.

Each of these pairs sums to (2), and they continue down to (0):

There are (54) positive integers from (1) to (54) and (53) negative integers from (-1) to (-53).

Step 3: Number of Pairs

The number of such pairs is:

[
\frac{53}{2} = 26.5 \text{ (this means we pair 26 times with one number left unpaired)}
]

Step 4: Calculate the Total Contribution from Pairs

The (26) pairs provide:

[
26 \times 2 = 52
]

And then we have the unpaired (0) which does not contribute anything additional. Now we need to also include (55) that was part of the first pair:

Adding this to our total gives us:

[
S = 52 + 55 = 107
]

Final Step: Sum

Therefore, the sum of the entire series from (-53) to (55) is:

[
\boxed{107}
]

13 Сен в 19:43
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