How to Form the Sum of the First n Terms of every Geometric Sequence or Geometric Series?

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 Sn and called the nth partial sum, can be found without having to add up all the terms. Recall that the first n terms of a geometric sequence are

a1,a1r,a1r2, …,a1r(n-2),a1r(n-1)

We proceed as follows:
is the sum of the first n terms of the sequence.
Sn=a1+a1r+a1r2+ ⋯ +a1r(n-2)+a1r(n-1)

Multiply both sides of the equation by r.
r⋅Sn=a1r+a1r2+a1r3+ ⋯ +a1r(n-1)+a1rn

Subtract the second equation from the first equation.
Snr⋅Sn=a1a1rn

Factor out Sn on the left and a1 on the right.
Sn (1-r)=a1⋅(1-rn)

Solve for Sn by dividing both sides by 1-r (assuming that r≠1).

Study Tip
If the common ratio is 1, the geometric sequence is

a1,a1,a1,a1, … .

The sum of the first n terms of this sequence is na1

We have proved the following result:

The Sum of the First n Terms of a Geometric Sequence
The sum, Sn of the first n terms of a geometric sequence is given by


in which a1 is the first term and r is 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:

1+2+4+8+16+ … +512

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:

-2S=-2-4-8- … -512-1024

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

If Sn=a1+a1r+a1r2+ ⋯ +a1r(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⋅Sn=-a1ra1r2a1r3– … –a1rn.

If we add Sn and –r⋅Sn, all but two of the terms on the right are eliminated:

Now divide each side of this equation by 1-r to get the formula for Sn.

Sum of n Terms of a Geometric Series
If Sn represents the sum of the first n terms of a geometric series with first term a1 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:

an=a1r(n-1)

where
n is the index of the sequence;
an is the nth-term of the sequence;
a1 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⋅SnSn=a1rna1
Sn (r-1)=a1⋅(rn-1)

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


Definition
A geometric series is 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 a1, a1r, a1r2, a1r3, … , a1r(n-1) is a geometric sequence with n terms, then

n(i=1)[a1r(i-1) ]=a1+a1r+a1r2+a1r3+ ⋯ +a1r(n-1)

is a geometric series.
Let the sum of these six terms be S6. Now multiply S6 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 S6.

Thus, S6=4,095. The pattern of a geometric series allows us to find a formula for the sum of the series. To Sn, add –r⋅Sn:


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
Sn=a+ar+ar2+ar3+ ⋯ +ar(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
Sn (1-r)=a(1-rn)

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 is a and the common ratio is r then the sum of the first n terms is


provided that r≠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 Sn, that is,

Sn=a+ar+ar2+ ⋯ +ar(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.

Snr⋅Sn=aarn
(1-r) Sn=a(1-rn)

Now, if r≠1 then

Thus, the sum to n terms of a geometric sequence is

Clearly Sn=na when r=1.


Geometric series
The formula is


where a is the first term, r is the common ratio, n is the number of terms, Sn is the sum of the terms.

Proof:
The general term of a geometric series is an=ar(n-1).

Sn=a1+a2+a3+ ⋯ +a(n-1)+an
Sn=a+ar+ar2+ ⋯ +ar(n-2)+ar(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 Sn and 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 Sn the sum to first n terms of geometric sequence Then

Sn=a+ar+ar2+ar3+ … +ar(n-1) … (1)

Case 1: If r=1,we have Sn=a+a+a+ … +a (n terms) =na
Case 2: If r≠1, multiplying (1) by r, we have
r⋅Sn=ar+ar2+ar3+ … +arn … (2)

Subtracting (2) from (1), we get (1-r) Sn=aarn. This gives


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
1+3+9+27+81+243+729+2187 or 3280!

The numbers 1, 3, 9, 27, 81, 243, 729, and 2187 form a geometric sequence in which a1=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 S8 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 sum Sn of the first n terms of a geometric series is given by


,where r≠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 with r=1, the terms are constant. For example, 4+4+4+ … +4 is such a series. In general, the sum of n terms of a geometric series with r=1 is na1.

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