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]
The series includes all integers from (-53) to (55).
To find how many terms are in this series, we can calculate:
The total count of integers from (-53) to (55) can be calculated by:
[\text{Total count} = 55 - (-53) + 1 = 55 + 53 + 1 = 109]
Now we can pair the terms in the series to simplify the addition. We can pair terms as follows:
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).
The number of such pairs is:
[\frac{53}{2} = 26.5 \text{ (this means we pair 26 times with one number left unpaired)}]
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]
Therefore, the sum of the entire series from (-53) to (55) is:
[\boxed{107}]
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:
[
Step 1: Count the NumbersS = -53 + (-52) + (-51) + \ldots + 0 + 1 + 2 + \ldots + 53 + 54 + 55
]
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:
[
Step 2: Pair the Terms\text{Total count} = 55 - (-53) + 1 = 55 + 53 + 1 = 109
]
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 PairsThe number of such pairs is:
[
Step 4: Calculate the Total Contribution from Pairs\frac{53}{2} = 26.5 \text{ (this means we pair 26 times with one number left unpaired)}
]
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:
[
Final Step: SumS = 52 + 55 = 107
]
Therefore, the sum of the entire series from (-53) to (55) is:
[
\boxed{107}
]