Formulas for Arithmetic Sequences and Series
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The origin of the formula to find a specific term of an arithmetic sequence where the common difference between terms is d can be seen by examining the sequence pattern.

Notice that the coefficient of d is one less than the location of the term.
Thus we have

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If we carry this idea further, we can find the formula for
partial sums of an arithmetic sequence (for the first n terns).

First examine the terms, as we did above, starting with the first term.

Remember:: a₁ + (n - 1) d = an

Now, try the same approach starting with the last term, and working backward.

This process of comparing a sequence forward and backward is credited to Carl Gauss.
Read about his story.

Now, add these two equations together and notice values that disappear.

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