How did we get these formulas?
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Arithmetic Formulas

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

If we carry this idea further, we can find the formula for partial sums of an arithmetic sequence.

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

Now, try the same approach starting with the last term.

Now, add these two equations and notice values that disappear.  We will end up with:

 

Geometric Formulas

The origin of the formula to find a specific term of an geometric sequence where the common ratio is r can also be seen by examining the sequence pattern.



Notice that the exponent of r is one less than the location of the term.
Thus we have

If we carry this idea further, we can find the formula for partial sums of an geometric sequence.

First express the series as we did above.

Now, multiply both sides by the common ratio, r.

Now, subtract these two equations and notice values that disappear.  We will end up with: