00:01
We have a sequence where the nth term is equal to negative 3 times 1 half to the n.
00:05
Let's prove that this is geometric.
00:07
Any geometric series will have a common ratio of r, where r is defined as the nth term divided by the nth, nth, nth, minus 1th term.
00:18
So with this in mind, let's plug in what we have.
00:20
Well, the nth term is going to be negative 3 times 1 half to the n, and the nth, n -minus -1th term will be negative 3 times 1 -half.
00:30
To the n minus 1.
00:32
We can simplify this.
00:33
There's a negative 3 on top and bottom, which can cancel out, leaving us with just one half to the n over one half to the n minus 1.
00:45
Now using our exponent rules, we can combine these terms.
00:48
That is, division is the same as subtracting exponents, leaving us with one half to the n minus n minus 1, subtracting the top exponent from the bottom.
00:59
Now, completing this subtraction will give us 1 half to the, well, n minus n is 0, and n minus negative 1, give us plus 1.
01:09
So 1ā2 to the first power, which is just equal to 1ā2, the common ratio...