**Use the formula for the sum of the first**

*n*terms of a geometric sequence## The Sum of the First *n* Terms of a Geometric Sequence

The sum of the first *n* terms of a geometric sequence, denoted by *S _{n}* and called the

**, can be found without having to add up all the terms. Recall that the first**

*n*th partial sum*n*terms of a geometric sequence are

*a*

_{1},

*a*

_{1}⋅

*r*,

*a*

_{1}⋅

*r*

^{2}, …,

*a*

_{1}⋅

*r*

^{(n-2)},

*a*

_{1}⋅

*r*

^{(n-1)}

We proceed as follows:

is the sum of the first

*n*terms of the sequence.

*S*=

_{n}*a*

_{1}+

*a*

_{1}⋅

*r*+

*a*

_{1}⋅

*r*

^{2}+ ⋯ +

*a*

_{1}⋅

*r*

^{(n-2)}+

*a*

_{1}⋅

*r*

^{(n-1)}

Multiply both sides of the equation by

*r*.

*r⋅S*=

_{n}*a*

_{1}⋅

*r*+

*a*

_{1}⋅

*r*

^{2}+

*a*

_{1}⋅

*r*

^{3}+ ⋯ +

*a*

_{1}⋅

*r*

^{(n-1)}+

*a*

_{1}⋅

*r*

^{n}Subtract the second equation from the first equation.

*S*–

_{n}*r⋅S*=

_{n}*a*

_{1}–

*a*

_{1}⋅

*r*

^{n}Factor out

*S*on the left and

_{n}*a*

_{1}on the right.

*S*(1-

_{n}*r*)=

*a*

_{1}⋅(1-

*r*)

^{n}Solve for

*S*by dividing both sides by 1-

_{n}*r*(assuming that

*r*≠1).

Study Tip

If the common ratio is 1, the geometric sequence is

a_{1},a_{1},a_{1},a_{1}, … .

The sum of the firstnterms of this sequence is na_{1}

We have proved the following result:

The Sum of the FirstnTerms of a Geometric Sequence

The sum,Sof the first_{n}nterms of a geometric sequence is given by

in whicha_{1}is the first term andris the common ratio (r≠1).

To find the sum of the terms of a geometric sequence, we need to know the first term, al, the common ratio, *r*, and the number of terms, *n*. The following examples illustrate how to use this formula.

**Finite Geometric Series**

Consider the following series:

The terms of this series are the terms of a finite geometric sequence. The indicated sum of a geometric sequence is called a

**geometric series**.

We can find the actual sum of this finite geometric series by using a technique similar to the one used for the sum of an arithmetic series. Let

*S*=1+2+4+8+ … +256+512.

Because the common ratio is 2, multiply each side by -2:

*S*=-2-4-8- … -512-1024

Adding the last two equations eliminates all but two of the terms on the right:

If

*S*=

_{n}*a*

_{1}+

*a*

_{1}⋅

*r*+

*a*

_{1}⋅

*r*

^{2}+ ⋯ +

*a*

_{1}⋅

*r*

^{(n-1)}is any geometric series, we can find the sum in the same manner. Multiplying each side of this equation by -r yields

*r⋅S*=-

_{n}*a*

_{1}⋅

*r*–

*a*

_{1}⋅

*r*

^{2}–

*a*

_{1}⋅

*r*

^{3}– … –

*a*

_{1}⋅

*r*.

^{n}If we add

*S*and –

_{n}*r⋅S*, all but two of the terms on the right are eliminated:

_{n}Now divide each side of this equation by 1-

*r*to get the formula for

*S*.

_{n}

Sum ofnTerms of a Geometric Series

IfSrepresents the sum of the first_{n}nterms of a geometric series with first terma_{1}and common ratio (r≠1), then

**Finite Geometric Series**

When we sum a known number of terms in a geometric sequence, we get a finite geometric series. We know that we can write out each term of a geometric sequence in the general form:

*a*=

_{n}*a*

_{1}⋅

*r*

^{(n-1)}

where

●

*n*is the index of the sequence;

●

*a*is the

_{n}*n*th-term of the sequence;

●

*a*

_{1}is the first term;

●

*r*is the common ratio (the ratio of any term to the previous term).

By simply adding together the first *n* terms, we are actually writing out the series

We may multiply the above equation by

*r*on both sides, giving us

You may notice that all the terms on the right side of (i) and (ii) are the same, except the first and last terms. If we subtract (i) from (ii), we are left with just

*r⋅S*–

_{n}*S*=

_{n}*a*

_{1}⋅

*r*–

^{n}*a*

_{1}

*S*(

_{n}*r*-1)=

*a*

_{1}⋅(

*r*-1)

^{n}Dividing by (

*r*-1) on both sides, we arrive at the general form of a geometric series:

Definition

Ageometric seriesis the indicated sum of the terms of a geometric sequence.

For example, 3, 12, 48, 192, 768, 3,072 is a geometric sequence with *r*=4. The indicated sum of this sequence, 3+12+48+192+768+3,072, is a geometric series.

In general, if *a*_{1}, *a*_{1}⋅*r*, *a*_{1}⋅*r*^{2}, *a*_{1}⋅*r*^{3}, … , *a*_{1}⋅*r*^{(n-1)} is a geometric sequence with *n* terms, then

^{n}

_{(i=1)}[

*a*

_{1}⋅

*r*

^{(i-1)}]=

*a*

_{1}+

*a*

_{1}⋅

*r*+

*a*

_{1}⋅

*r*

^{2}+

*a*

_{1}⋅

*r*

^{3}+ ⋯ +

*a*

_{1}⋅

*r*

^{(n-1)}

is a geometric series.

Let the sum of these six terms be

*S*

_{6}. Now multiply

*S*

_{6}by the negative of the common ratio, -4. This will result in a series in which each term but the last is the

*opposite*of a term in

*S*

_{6}.

Thus,

*S*

_{6}=4,095. The pattern of a geometric series allows us to find a formula for the sum of the series. To

*S*, add –

_{n}*r⋅S*:

_{n}**The sum of a geometric series**

Suppose that we want to find the sum of the first *n* terms of a geometric sequence. What we get is

*S _{n}*=

*a*+

*a*⋅

*r*+

*a*⋅

*r*

^{2}+

*a*⋅

*r*

^{3}+ ⋯ +

*a*⋅

*r*

^{(n-1)},

and this is called a

*geometric series*. Now the trick here to find the sum is to multiply by

*r*and then subtract:

so that

*S*(1-

_{n}*r*)=

*a*(1-

*r*)

^{n}Now divide by 1-

*r*(as long as

*r*≠1) to give

Key Point

The sum of the terms of a geometric sequence gives a geometric series. If the starting value isaand the common ratio isrthen the sum of the firstnterms is

provided thatr≠1.

**Sum to n Terms of a Geometric Sequence**

We wish to find the sum of first *n* terms of the geometric sequence whose first term is *a* and the common ratio *r*.

Let us denote the sum of first 11 terms by *S _{n}*, that is,

*S*=

_{n}*a*+

*a*⋅

*r*+

*a*⋅

*r*

^{2}+ ⋯ +

*a*⋅

*r*

^{(n-1)}

Let us multiply both sides of this equation by

*r*. We write the results of this operation below the original equation and line up vertically the terms of the same exponent (to prepare for a subtraction).

Now subtract both the sides of lower equation [equation (2)] from the upper equation [equation (1)].

Most of the summands on the right side cancel. All that is left is the equation.

*S*–

_{n}*r⋅S*=

_{n}*a*–

*a*⋅

*r*

^{n}(1-

*r*)

*S*=

_{n}*a*(1-

*r*)

^{n}Now, if

*r*≠1 then

Thus, the sum to

*n*terms of a geometric sequence is

Clearly

*S*=na when

_{n}*r*=1.

**Geometric series**

The formula is

where

*a*is the first term,

*r*is the common ratio,

*n*is the number of terms,

*S*is the sum of the terms.

_{n}**Proof**:

The general term of a geometric series is *a _{n}*=

*a*⋅

*r*

^{(n-1)}.

*S*=

_{n}*a*

_{1}+

*a*

_{2}+

*a*

_{3}+ ⋯ +

*a*

_{(n-1)}+

*a*

_{n}*S*=

_{n}*a*+

*a*⋅

*r*+

*a*⋅

*r*

^{2}+ ⋯ +

*a*⋅

*r*

^{(n-2)}+

*a*⋅

*r*

^{(n-1)}

Multiply each term by

*r*, write down the series again with like terms under each other. Subtract each bottom term from each top term

*S*and

_{n}*a*are common factors.

We can also use for

The proof for must be learnt for exams.

**Sum to n terms of a Geometric Sequence**

Let the first term of a geometric sequence be *a* and the common ratio be *r*. Let us denote by *S _{n}* the sum to first

*n*terms of geometric sequence Then

*S*=

_{n}*a*+

*a*⋅

*r*+

*a*⋅

*r*

^{2}+

*a*⋅

*r*

^{3}+ … +

*a*⋅

*r*

^{(n-1)}… (1)

Case 1: If

*r*=1,we have

*S*=

_{n}*a*+

*a*+

*a*+ … +

*a*(

*n*terms) =na

Case 2: If

*r*≠1, multiplying (1) by

*r*, we have

*r⋅S*=

_{n}*a*⋅

*r*+

*a*⋅

*r*

^{2}+

*a*⋅

*r*

^{3}+ … +

*a*⋅

*r*… (2)

^{n}Subtracting (2) from (1), we get (1-

*r*)

*S*=

_{n}*a*–

*a*⋅

*r*. This gives

^{n}Is e-mailing a joke like a geometric series?

Suppose you e-mail a joke to three friends on Monday. Each of those friends sends the joke on to three of their friends on Tuesday. Each person who receives the joke on Tuesday sends it to three more people on Wednesday, and so on.

GEOMETRIC SERIES

Notice that every day, the number of people who read your joke is three times the number that read it the day before. By Sunday, the number of people, including yourself, who have read the joke is

The numbers 1, 3, 9, 27, 81, 243, 729, and 2187 form a geometric sequence in which *a*_{1}=1 and *r*=3. Since 1, 3, 9, 27, 81, 243, 729, 2187 is a geometric sequence, 1+3+9+27+81+243+729+2187 is called a geometric series. Below are some more examples of geometric sequences and their corresponding geometric series.

To develop a formula for the sum of a geometric series, consider the series given in the e-mail situation above.

The expression for

*S*

_{8}can be written as

A rational expression like this can be used to find the sum of any geometric series.

Key Concept:Sum of a Geometric Series

The sumSof the first_{n}nterms of a geometric series is given by

,wherer≠1.

You cannot use the formula for the sum with a geometric series for which

r=1 because division by 0 would result. In a geometric series withr=1, the terms are constant. For example, 4+4+4+ … +4 is such a series. In general, the sum ofnterms of a geometric series withr=1 isn⋅a_{1}.