Prove the formula for the nth term of a geometric sequence.

The formula for the nth term of a geometric sequence is an = a1 * r^(n-1).

To prove this formula, we can use mathematical induction. First, we need to show that the formula holds for n = 1. In a geometric sequence, the first term is a1 and the common ratio is r. Therefore, when n = 1, the nth term is simply a1 * r^(1-1) = a1, which is the first term of the sequence. So, the formula holds for n = 1.

Next, we assume that the formula holds for some arbitrary value of n, say k. That is, ak = a1 * r^(k-1). We need to show that the formula also holds for n = k+1. The (k+1)th term of the sequence is given by ak+1 = ak * r. Substituting the value of ak from our assumption, we get:

ak+1 = a1 * r^(k-1) * r
= a1 * r^k

This is exactly the formula for the (k+1)th term that we wanted to prove. Therefore, by mathematical induction, the formula an = a1 * r^(n-1) holds for all positive integers n.

Another way to prove the formula is by using the formula for the sum of a geometric series. The sum of the first n terms of a geometric sequence is given by:

Sn = a1 * (1 - r^n) / (1 - r)

We can rearrange this formula to get:

a1 = Sn * (1 - r) / (1 - r^n)

Substituting this value of a1 in the formula for the nth term, we get:

an = Sn * (1 - r) / (1 - r^n) * r^(n-1)

Simplifying this expression, we get:

an = Sn * (1 - r^(n-1))

Using the formula for Sn, we can rewrite this as:

an = a1 * r^(n-1)

Therefore, the formula for the nth term of a geometric sequence is proved.