WEBVTT
00:00:00.860 --> 00:00:06.270
Find the sum of the geometric series thirteen halves plus thirteen fourths plus thirteen eighths.
00:00:07.040 --> 00:00:19.040
The formula for a sum that’s going to infinity is 𝑎 divided by one minus 𝑟, where 𝑎 is the initial amount, and in this case that will be thirteen halves, and 𝑟 is the common ratio.
00:00:19.580 --> 00:00:24.420
The common ratio is the number that we multiply to each term to get the following term.
00:00:24.890 --> 00:00:26.990
So we can solve for 𝑟 by dividing.
00:00:27.520 --> 00:00:36.990
If we would take thirteen fourths and divide by thirteen halves, that would tell us what we multiply thirteen halves by in order to get thirteen fourths.
00:00:37.790 --> 00:00:43.380
However, when we divide fractions, we multiply by the reciprocal.
00:00:43.750 --> 00:00:46.180
So instead of dividing, we multiply.
00:00:46.320 --> 00:00:49.880
And we multiply by the reciprocal, so we flip our second fraction.
00:00:50.390 --> 00:00:59.900
We can multiply straight across and get twenty-six fifty seconds and then reduce or cancel the thirteenths and two goes into four twice.
00:00:59.900 --> 00:01:02.920
So on the numerator, there’s nothing, so that means there’s really a one.
00:01:03.090 --> 00:01:05.280
And on the denominator, there’s just a two.
00:01:05.950 --> 00:01:09.380
So 𝑟 is equal to one-half, and now we can play again.
00:01:09.920 --> 00:01:13.750
So we have thirteen halves divided by one minus one-half.
00:01:14.580 --> 00:01:16.420
So let’s first begin with the denominator.
00:01:16.450 --> 00:01:19.630
One minus one-half is just one-half now.
00:01:19.630 --> 00:01:27.550
Now, that’s a pretty easy fraction to work with, but just a kind of review, when we add and subtract fractions, we need common denominators.
00:01:27.580 --> 00:01:35.370
So we could change one into two over two and we subtract the numerators and keep our denominator, which again is one-half.
00:01:35.400 --> 00:01:42.380
Two minus one is one and then we keep the two on the bottom, so thirteen halves divided by one-half, which we could rewrite like this.
00:01:42.570 --> 00:01:44.490
And then remember we flip and multiply.
00:01:44.800 --> 00:01:45.880
The twos cancel.
00:01:45.880 --> 00:01:49.180
So this means the sum of our geometric series is 13.