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Ever heard the story about Carl Friedrich Gauss, the math genius? When he was in elementary school, his teacher assigned the task of summing all numbers from to : . The teacher hoped this would keep the students busy for a while.
But Gauss had a brilliant idea! He didn't sum them one by one. This sequential summation of terms from an arithmetic sequence (a sequence with a constant difference between terms) is what we call an Arithmetic Series.
For example, is an arithmetic sequence with the first term and a common difference . The corresponding arithmetic series is .
Gauss noticed an interesting pattern:
It turns out there are pairs of numbers, each summing to . So, the total sum is . Clever, right?
We can use Gauss's method to derive a general formula for the sum of the first terms of an arithmetic series, usually denoted by .
Let's say we have an arithmetic series:
If written out with the first term and the common difference :
Now, let's rewrite the series in reverse order, from the last term to the first:
Or:
Next, let's add these two versions of together, term by term:
Notice! The sum of each pair of terms (top and bottom) is always the same, which is . Since there are terms, there are such identical sums.
So, we get:
By dividing both sides by , we obtain the formula for the sum of the first terms of an arithmetic series:
There are two main formulas commonly used to calculate :
If the first term and the common difference are known:
If the first term and the -th term are known: Recall the formula for the -th term is . Substituting this into the first formula:
Notation:
Recalculate the sum of the series .
Given:
Since and are known, we use the second formula:
The result is exactly the same as Gauss's calculation!
Given the arithmetic series: . Calculate the sum of the first terms !
Given:
Since and are known, we use the first formula:
So, the sum of the first terms of this series is .
This second formula resembles Gauss's method: the sum of the first and last terms, multiplied by the number of pairs .
