### Video Transcript

In this video, we will learn how to
calculate the sum of the terms in a geometric sequence with a finite number of
terms. We will begin by recalling what we
mean by a finite geometric sequence. In a geometric sequence, each term
is found by multiplying the previous term by a constant, for example, the sequence
two, six, 18, 54, and so on. To get from the first to the second
term, second to the third term, and third to the fourth term, we need to multiply by
three, as two multiplied by three is six, six multiplied by three is 18, and 18
multiplied by three is 54.

This constant in the example three
is known as the common ratio denoted by the letter π. We let the first term of any
geometric sequence be the letter π or π sub one. As we multiply this by the constant
π to get the next term, the second term, π sub two, is equal to ππ. The third term is equal to ππ
squared. This pattern continues such that
the πth term is equal to π multiplied by π to the power of π minus one. This gives us a general formula for
the πth term of a geometric sequence. We will now look at how we can
calculate the sum of a finite geometric sequence.

The sum of the first π-terms of a
geometric sequence can be written as follows. π sub π is equal to π plus ππ
plus ππ squared, and so on, up to the last two terms of π multiplied by π to the
power of π minus two and π multiplied by π to the power of π minus one. We will call this equation one. If we multiply each of the terms in
this equation by π, we get π multiplied by π sub π is equal to ππ plus ππ
squared plus ππ cubed, and so on, such that the last two terms are π multiplied
by π to the power of π minus one and π multiplied by π to the power of π. We will call this equation two and
then subtract this equation from equation one.

On the left-hand side, we have π
sub π minus π multiplied by π sub π. We can factor out π sub π such
that this becomes π sub π multiplied by one minus π. On the right-hand side, when
subtracting, the ππ terms will cancel, likewise, ππ squared. In fact, all the terms will cancel
apart from π in equation one and π multiplied by π to the power of π in equation
two. This means that the right-hand side
becomes π minus π multiplied by π to the power of π. These two terms have a common
factor of π, so we can factor this out.

Next, we can divide both sides of
our equation by one minus π. This gives us π sub π is equal to
π multiplied by one minus π to the power of π all divided by one minus π. This formula enables us to
calculate the sum of the first π-terms of a finite geometric sequence. We will now look at some questions
where we need to apply this formula.

A geometric sequence has a first
term of three and a common ratio of five. Find the sum of the first six
terms.

We know that the sum of the first
π-terms of a geometric sequence, written π sub π, is equal to π multiplied by
one minus π to the power of π all divided by one minus π. In this question, we are told that
the first term π is equal to three. The common ratio π is equal to
five. And we are interested in the sum of
the first six terms. Therefore, π is equal to six. Substituting in these values, we
have that π sub six is equal to three multiplied by one minus five to the power of
six all divided by one minus five. Five to the power of six or five to
the sixth power is equal to 15625 and one minus five is equal to negative four. Typing this calculation into our
calculator gives us 11718. The sum of the first six terms of
the geometric sequence with first term three and common ratio five is 11718.

In our next question, we will need
to calculate the common ratio π and the number of terms π before calculating the
sum of the sequence.

Find the sum of the geometric
sequence 16, negative 32, 64, and so on, all the way up to 256.

We know that the sum of any
geometric sequence denoted π sub π is equal to π multiplied by one minus π to
the power of π all divided by one minus π. We can see immediately from the
sequence that the value of the first term π is 16. The second term is equal to π
multiplied by π, and this is equal to negative 32. If we label these equations one and
two, we can calculate the value of π by dividing equation two by equation one. On the left-hand side, we have ππ
divided by π, and on the right-hand side negative 32 divided by 16. As π is not equal to zero, we can
cancel this on the left-hand side. And negative 32 divided by 16 is
negative two.

This value of π makes sense as we
multiply the first term 16 by negative two to get the second term negative 32. This also works to get from the
second to third term. Negative 32 multiplied by negative
two is 64. We know that the πth term of any
geometric sequence written π sub π is equal to π multiplied by π to the power of
π minus one. To calculate the value of π, we
can substitute our values of π and π and the πth term 256. This gives us the equation 256 is
equal to 16 multiplied by negative two to the power of π minus one. We can divide both sides by 16 such
that 16 is equal to negative two to the power of π minus one.

We know that negative two to the
fourth power or negative two to the power of four is equal to 16. This means that π minus one must
be equal to four. Adding one to both sides of this
equation gives us a value of π equal to five. We now have values of π, π, and
π. The sum of the first five terms is
therefore equal to 16 multiplied by one minus negative two to the fifth power all
divided by one minus negative two. This simplifies to 16 multiplied by
one plus 32 all divided by three. Typing this into the calculator, we
get an answer of 176. The sum of the geometric sequence
16, negative 32, 64, and so on, up to 256 is equal to 176.

In our next question, we need to
calculate the number of terms in a geometric sequence.

The number of terms of a geometric
sequence whose first term is 729, last term is one, and sum of all terms is 1093 is
blank.

Weβre told that the first term of
our sequence π sub one or π is 729. The last term π sub π is equal to
one. We know that π sub π is equal to
π multiplied by π to the power of π minus one. We are also told that the sum of
all the terms π sub π is equal to 1093, where π sub π is equal to π multiplied
by one minus π to the power of π all divided by one minus π. Our aim in this question is to
calculate the number of terms π.

Substituting in the value of π, we
see that 729 multiplied by π to the power of π minus one is equal to one. Dividing both sides of this
equation by 729, π to the power of π minus one is equal to one over 729. Using one of our laws of exponents
or indices, we can rewrite the left-hand side as π to the power of π over π to
the power of one, as π₯ to the power of π minus π is equal to π₯ to the power of
π divided by π₯ to the power of π. We can then multiply both sides of
this equation by π, giving us π sub π is equal to one over 729 π.

We will now clear some space and
consider the second formula. This gives us 729 multiplied by one
minus one over 729 π all divided by one minus π is equal to 1093. We can distribute the parentheses
of the numerator on the left-hand side to give us 729 minus π. Multiplying through by one minus π
gives us 729 minus π is equal to 1093 multiplied by one minus π. We can once again distribute the
parentheses or expand the brackets giving us 1093 minus 1093π. Subtracting 729 and adding 1093π
to both sides gives us 1092π is equal to 364. We can then divide both sides by
1092 such that π is equal to one-third. We can now substitute this back in
to calculate the value of π.

One-third to the power of π is
equal to one over 729 multiplied by one-third. We know that three to the power of
six is equal to 729. This means that one-third to the
power of six is equal to one over 729. This means that we can rewrite the
right-hand side of our equation as one-third to the power of six multiplied by
one-third. Once again, using our laws of
exponents, we can add the powers. Six plus one is equal to seven. As one-third to the power of π is
equal to one-third to the power of seven, then π must be equal to seven. The number of terms of the
geometric sequence whose first term is 729, last term is one, and sum of all terms
is 1093 is seven.

In our final question, we will once
again need to find the sum of π-terms of a geometric sequence.

Find the sum of the first seven
terms of a geometric sequence given π sub five is equal to negative eight
multiplied by π sub two and π sub four plus π sub six is equal to negative
64.

We know that the πth term of any
geometric sequence denoted π sub π is equal to π multiplied by π to the power of
π minus one. The fifth term of the sequence will
therefore be equal to π multiplied by π to the fourth power, and the second term
π sub two will be equal to ππ. Likewise, the fourth term will be
equal to ππ cubed, and the sixth term π sub six will be equal to π multiplied by
π to the fifth power. We can therefore rewrite our two
equations.

Firstly, we have π multiplied by
π to the fourth power is equal to negative eight ππ. As both π and π cannot be equal
to zero, we can divide through by π and π. This leaves us with π cubed is
equal to negative eight. We can cube root both sides of this
equation such that π is equal to negative two. We can then substitute this value
of π into our second equation. ππ cubed plus ππ to the fifth
power is equal to negative 64. Negative two cubed is equal to
negative eight. So the first term becomes negative
eight π. Negative two to the fifth power is
equal to negative 32. Our equation becomes negative eight
π plus negative 32π is equal to negative 64.

The left-hand side simplifies to
negative 40π. We can then divide through by
negative 40 such that π is equal to eight over five or eight-fifths. This is also equal to the decimal
1.6. We now have a value of π, a value
of π, and a value of π equal to seven, as we need to calculate the sum of the
first seven terms. In order to do this, we will use
the formula π sub π is equal to π multiplied by one minus π to the power of π
all divided by one minus π. π sub seven is therefore equal to
1.6 or eight over five multiplied by one minus negative two to the seventh power all
divided by one minus negative two. Typing this into the calculator
gives us an answer of 344 over five. As a decimal, this is equal to
68.8. The sum of the first seven terms of
a geometric sequence given that π sub five is equal to negative eight multiplied by
π sub two and π sub four plus π sub six is equal to negative 64 is 344 over five
or 68.8.

We will now summarize the key
points from this video. A geometric sequence has a first
term π and a common ratio π. And each term is found by
multiplying the previous term by this common ratio. The πth term of any geometric
sequence written π sub π is equal to π multiplied by π to the power of π minus
one. And the sum of the first π-terms
of a geometric sequence written π sub π is equal to π multiplied by one minus π
to the πth power divided by one minus π. We saw in this video that we can
use these formulae to calculate the sum of the terms in a finite geometric
sequence.