### Video Transcript

Find the sum of the first six terms
of the geometric series a half plus a quarter plus an eighth plus a 16th and so
on.

So the first thing to do in this
type of questions is that actually letβs think what is a geometric series? Well, a geometric series is
actually a series with a common ratio between successive terms. So that actually means that if I
divide a term by the previous term, itβs going to give us the same value each
time.

Okay, great, so now we know what it
is. But how do we actually work out the
sum of the first six terms? Well, to work out the sum of any
number of terms, we actually have a formula. And the formula is that the sum of
the number of terms β so π π β is equal to π and then one minus π to the power
of π divided by one minus π. And this is when π is equal to the
first term and π is equal to the common ratio.

Okay, so we actually have this
formula. Letβs use it to find the sum of our
first six terms of the geometric series. Well, the first thing we need to do
is actually find out what π and π are. Well, π is going to be a half
because itβs our first term. So weβve got π. Now what π going to be? Well, as we said, π is gonna be
our common ratio. So what we need to do actually to
find out our common ratio is actually find a term and then divide it by the previous
term. And that will give us our common
ratio.

So Iβm gonna take our second and
first terms. So weβre gonna say that π is equal
to a quarter which is our second term divided by our first term which is a half. And then, we can say this is equal
to a quarter multiplied by two over one because remember if weβre dividing by a
fraction, weβre actually find the reciprocal and multiply. So therefore, we can say that π is
gonna be equal to two over four which is equal to a half.

Okay, great, weβve now found our
common ratio. Okay, now, we found our π and
π. There is one more variable that we
need to know before we substitute into our formula. And thatβs π. Well, π we said was actually the
number of terms. And if we look back at the
question, we can see that- ok, well, the number of terms is gonna be six because we
want to find the sum of the first six terms.

So now weβve got all the variables
we need to actually substitute back into the formula. So letβs do that and find the sum
of our first six terms. So weβre gonna get the sum of the
first six terms is equal to a half multiplied by one minus a half to the power of
six divided by one minus a half which is gonna be equal to a half multiplied by one
minus one over 64 and then divided by a half. Well, therefore, we can actually
divide top and bottom β so the numerator and the denominator by a half.

So therefore, we can say that the
sum of the first six terms is gonna be equal to 63 over 64. And we got that because if you have
one minus one over 64, that will be like 64 over 64 minus one over 64, which gives
us 63 over 64.