00:01
We have a sequence where the nth term is equal to five halves to the n power.
00:04
Let's prove that it's geometric.
00:06
Now, any geometric sequence has a common ratio of r, where r is defined as the nth term, and is bn over the n minus 1th term, bn minus 1.
00:19
Okay, so with that in mind, let's plug these things in and find the common ratio, if it exists.
00:24
So the nth term, that's going to be five halves to the n, and the n minus 1th term will be five halves to the n minus 1.
00:34
Now, we can simplify this a little bit.
00:37
As we have the same thing divided on top and bottom, we can use an exponent rule, which states that with division, you can rewrite it as subtraction of exponents.
00:46
That is, this will be five halves to the n minus n minus one.
00:52
That is top exponent minus bottom.
00:55
Well, this is equal to five halves.
00:57
So we have n minus n minus 1, the ends will cancel out, and the negatives will cancel on the 1, leaving us with just the first power.
01:07
And anything of the first power remains the same, so we just have five halves.
01:12
Thus, that is our common ratio, meaning that this is, in fact, geometric.
01:16
Now it's time for the second part...