We want to find the partial sum
for the geometric sequence.
Since we are summing a sequence,
the directions could also say
just find the sum of the geometric series.
Since we know we have a geometric series,
we can use this partial sum formula here,
where s sub n is equal to a sub one times the quantity
one minus r raised to the power of n divided by the quantity
one minus r,
where a sub one is the first term, r is the common ratio,
and n would be the number of terms.
So, looking at our series, the first term is negative two,
so we know a sub one is equal to negative two.
The common ratio r would be the constant
that we multiply it by to find successive terms.
Notice that they multiply it by two,
negative two times two is negative four,
negative four times two is negative eight, and so on,
so our common ratio r is equal to positive two.
They weren't able to determine r by analyzing the pattern
because we know we're summing a geometric sequence,
we can use this formula here to determine r,
where r is equal to a sub n divided by a sub n minus one,
which means if select a term
and divide by the term before it,
we can also find our common ratio r.
Notice that negative four divided by negative two
is positive two.
Negative eight divided by negative four is positive two,
and so on.
But notice, these are partial sum formula.
We also have to find n,
which would be the number of terms in this series.
And again, since we know that the terms
come from a geometric sequence,
we can use the formula a sub n is equal to
a sub one times r, raised to the power of n minus one,
to determine the value of n.
So, let's go ahead and do that.
We're going to let a sub n be equal to the last term,
so we can find the value of n,
so that would give us the equation negative 64
equals a sub one, the first term, which is negative two,
times r which we know is positive two
to the power of n minus one.
Let's go ahead and isolate this exponential part
by dividing both side by negative two.
This would give us 32 is equal to two
raised to the power of n minus one.
And now, to solve for n, we'll obtain a common base,
and therefore, the exponents must be equal.
32 is equal to five factors of two,
so we can write this as two to the fifth
is equal to two to the power of n minus one.
Well, if these are equal and the bases are equal,
then five must equal n minus one.
Five equals n minus one.
Add one on both sides, we have n equal six.
So this series has six terms,
so now we have all the information we need
to find the partial sum.
Since n is six, we have s sub six equals
a sub one, which is negative two,
times the quantity one minus two raised to the sixth power,
divided by one minus r, which again, is two.
Now we'll go ahead and simplify this
to find our partial sum.
We have one minus two to the sixth,
where two to the sixth is equal to 64.
One minus 64 is equal to negative 63.
One minus two is equal to negative one,
so we have negative two times negative 63,
that's positive 126, divided by negative one,
so we have negative 126 as our partial sum.
Since this series only has six terms,
since n is equal to six, and we already have four of them,
let's go ahead and find the remaining two,
to check this the long way to actually find the sum
rather than using the formula.
Since r is equal to two, we'll multiply it by two
to find the next term.
Negative eight times two is negative 16.
Negative 16 times two is equal to negative 32.
Negative 32 times two gives us our last term of negative 64.
And the sum is negative 126,
verifying that we applied our partial sum formula correctly.
I hope you have found this explanation helpful.