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Prove the formula for the sum of a finite geometric series.

The formula for the sum of a finite geometric series is Sn = a(1-r^n)/(1-r).

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant factor. The sum of a finite geometric series is given by the formula Sn = a(1-r^n)/(1-r), where a is the first term, r is the common ratio, and n is the number of terms.

To prove this formula, we can use the method of partial sums. Let S1 be the sum of the first term, S2 be the sum of the first two terms, and so on, up to Sn, which is the sum of the first n terms. Then we have:

S1 = a
S2 = a + ar
S3 = a + ar + ar^2
...
Sn = a + ar + ar^2 + ... + ar^(n-1)

To find a formula for Sn, we can subtract S(n-1) from Sn:

Sn - S(n-1) = ar^(n-1)

We can simplify this expression by multiplying both sides by (1-r):

Sn(1-r) - S(n-1)(1-r) = ar^(n-1)(1-r)

Using the formula for the sum of a geometric series, we can simplify the left-hand side:

Sn(1-r) - S(n-1)(1-r) = a(1-r^n)

Dividing both sides by (1-r), we get:

Sn = a(1-r^n)/(1-r)

This proves the formula for the sum of a finite geometric series.

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