Lesson Explainer: Sum of a Finite Geometric Sequence Mathematics

In this explainer, we will learn how to calculate the sum of the terms in a geometric sequence with a finite number of terms.

Let us consider a sequence where each term is found by multiplying the previous term by a constant, for example, 2,6,18,54,162,486,โ€ฆ.

We call this constant multiplier the common ratio. Another way to describe the sequence would be to say that each term in the sequence is equal to the previous term multiplied by the common ratio.

This is known as a geometric sequence, in this case with a first term equal to 2 and a common ratio of 3. If our sequence just consisted of, say, the six terms above (or indeed any specific number of terms), then we would call it a finite geometric sequence because it has a finite number of terms. If the sequence followed this pattern forever, as the ellipsis at the end implies, then we would call it an infinite geometric sequence.

Definition: Geometric Sequence

A geometric sequence is a sequence that has a common ratio, ๐‘Ÿ, between consecutive terms. The first term is denoted by ๐‘‡ or ๐‘‡๏Šง, the second term ๐‘‡๏Šจ, the third term ๐‘‡๏Šฉ, and so on. The ๐‘›th term is denoted by ๐‘‡๏Š.

Each term is equal to the previous term multiplied by the common ratio: ๐‘‡=๐‘‡,๐‘‡=๐‘‡ร—๐‘Ÿ,๐‘‡=๐‘‡ร—๐‘Ÿ,โ€ฆ,๐‘‡=๐‘‡ร—๐‘Ÿ.๏Šง๏Šจ๏Šง๏Šฉ๏Šจ๏Š๏Š๏Šฑ๏Šง

This can also be expressed as the first term multiplied by powers of the common ratio: ๐‘‡=๐‘‡,๐‘‡=๐‘‡๐‘Ÿ,๐‘‡=๐‘‡๐‘Ÿ,โ€ฆ,๐‘‡=๐‘‡๐‘Ÿ.๏Šง๏Šจ๏Šฉ๏Šจ๏Š๏Š๏Šฑ๏Šง

Returning to our initial geometric sequence above, if we know the numbers in the sequence, we can calculate the common ratio by dividing the value of one term by the value of the previous term. Since the ratio is common between all successive pairs of terms, it does not matter which pair we choose for our calculation.

The ratio of the first two terms is 6รท2=3, the ratio of the second two terms is 18รท6=3, and so on.

Definition: The Common Ratio

Since we multiply one term by the common ratio to get the next term, we can express this generally as ๐‘‡=๐‘‡ร—๐‘Ÿ,๏Š๏Šฐ๏Šง๏Š and by dividing both sides of the equation by ๐‘‡๏Š, we get ๐‘Ÿ=๐‘‡๐‘‡.๏Š๏Šฐ๏Šง๏Š

Alternatively, with the definition that one term is the result of multiplying the previous term by the common ration, we find that ๐‘Ÿ=๐‘‡๐‘‡.๏Š๏Š๏Šฑ๏Šง

The sum of the terms in a sequence is called a series. Given that a geometric sequence is 2,6,18,54,162,486,1458,4374,13122,39366โ€ฆ, the corresponding geometric series could be represented as shown below: 2+6+18+54+162+486+1458+4374+13122+39366โ€ฆ.

In this case, by adding together the first 10 terms in the series, we can see that the sum of these terms is 59โ€Žโ€‰โ€Ž048.

We will now derive a formula for the sum of the first ๐‘› terms of a geometric sequence.

Consider a geometric sequence with first term ๐‘‡ and common ratio ๐‘Ÿ. The first ๐‘› terms can be written as ๐‘‡,๐‘‡๐‘Ÿ,๐‘‡๐‘Ÿ,โ€ฆ,๐‘‡๐‘Ÿ๏Šจ๏Š๏Šฑ๏Šง, so the sum of the first ๐‘› terms of a geometric sequence can be written as follows:

๐‘†=๐‘‡+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๏Š๏Šจ๏Š๏Šฑ๏Šจ๏Š๏Šฑ๏Šง(1)

If we multiply both sides of our equation by ๐‘Ÿ, we have

๐‘Ÿ๐‘†=๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๏Š๏Šจ๏Šฉ๏Š๏Šฑ๏Šง๏Š(2)

Subtracting equation (2) from equation (1), all the terms on the right-hand side apart from ๐‘‡ and ๐‘‡๐‘Ÿ๏Š cancel out. ๐‘†=๐‘‡+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๐‘Ÿ๐‘†=๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๏Š๏Šจ๏Š๏Šฑ๏Šจ๏Š๏Šฑ๏Šง๏Š๏Šจ๏Šฉ๏Š๏Šฑ๏Šง๏Š

So, ๐‘†โˆ’๐‘Ÿ๐‘†=๐‘‡โˆ’๐‘‡๐‘Ÿ.๏Š๏Š๏Š

Factoring ๐‘†๏Š from the right-hand side and ๐‘‡ from the left-hand side will allow us to create an equation for ๐‘†๏Š: ๐‘†(1โˆ’๐‘Ÿ)=๐‘‡(1โˆ’๐‘Ÿ)๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ.๏Š๏Š๏Š๏Š

Alternatively, we could have subtracted (1) from equation (2) to obtain the formula ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š

Definition: The Sum of a Finite Geometric Sequence

The sum of the first ๐‘› terms of a geometric sequence, with first term ๐‘‡ and common ratio ๐‘Ÿ, is denoted by ๐‘†๏Š: ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š๏Š๏Šor

Generally, we use the first version when ๐‘Ÿ<1 and the second one when ๐‘Ÿ>1.

If ๐‘Ÿ=1, all the terms of the geometric sequence are the same, so we would just need to multiply the first term by the number of terms: ๐‘†=๐‘‡ร—๐‘›๏Š.

In our first example, we will calculate the sum of the first six terms of a geometric sequence, given its common ratio and first term.

Example 1: Calculating the Sum of Certain Terms in a Finite Geometric Series

A geometric series has a first term of 3 and a common ratio of 5. Find the sum of the first 6 terms.

Answer

In order to answer this question, we will use the formula to calculate the sum of the first ๐‘› terms of a geometric sequence, with first term ๐‘‡ and common ratio ๐‘Ÿ: ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š

We are told in the question that the first term is 3, the common ratio is 5, and we are calculating the sum of the first 6 terms; so, let ๐‘‡=3, ๐‘Ÿ=5, and ๐‘›=6: ๐‘†=3๏€น5โˆ’1๏…5โˆ’1๐‘†=3(15624)4๐‘†=11718.๏Šฌ๏Šฌ๏Šฌ๏Šฌ

The sum of the first 6 terms of the geometric sequence is 11โ€Žโ€‰โ€Ž718.

In our second example, we will use the formula for the ๐‘›th term of a finite geometric sequence to help us calculate the number of terms and, hence, find the sum of the series.

Example 2: Finding the Sum of a Finite Geometric Sequence

Find the sum of the geometric sequence (16,โˆ’32,64,โ€ฆ,256).

Answer

We define the first term of a geometric sequence as ๐‘‡. Let ๐‘‡=16.

We can calculate the value of the common ratio, ๐‘Ÿ, by dividing any term by the term that precedes it: ๐‘Ÿ=โˆ’3216=โˆ’2๐‘Ÿ=64โˆ’32=โˆ’2.or

The common ratio of the sequence is equal to โˆ’2. Let ๐‘Ÿ=โˆ’2.

The last term of the sequence is 256. If we define the number of terms in the sequence to be ๐‘›, then we can say that ๐‘‡=256๏Š. The formula for the ๐‘›th term of a geometric sequence with first term ๐‘‡ and common ratio ๐‘Ÿ is given by ๐‘‡=๐‘‡๐‘Ÿ256=16(โˆ’2)16=(โˆ’2).๏Š๏Š๏Šฑ๏Šง๏Š๏Šฑ๏Šง๏Š๏Šฑ๏Šง

We know that (โˆ’2)=16.๏Šช

So, (โˆ’2)=(โˆ’2).๏Šช๏Š๏Šฑ๏Šง

Therefore, ๐‘›โˆ’1=4๐‘›=5.

To find the sum of the series, we can now use the formula ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๏Š๏Š, with ๐‘‡=16, ๐‘Ÿ=โ€“2, and ๐‘›=5: ๐‘†=16๏€บ1โˆ’(โˆ’2)๏†1โˆ’(โˆ’2)๐‘†=16(33)3๐‘†=176.๏Šซ๏Šซ๏Šซ๏Šซ

The sum of the geometric sequence 16,โˆ’32,64,โ€ฆ,256 is 176.

In our next example, we will need to rearrange our formulae to calculate the number of terms in a particular geometric sequence.

Example 3: Finding the Number of Terms of a Finite Geometric Sequence given Its Sum

The number of terms of a geometric sequence whose first term is 729, last term is 1, and sum of all terms is 1โ€Žโ€‰โ€Ž093 is .

Answer

We define the first term of a geometric sequence as ๐‘‡. Let ๐‘‡=729.

The last term ๐‘‡=1๏Š, where ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง and ๐‘Ÿ is the common ratio between terms: 1=729๐‘Ÿ1729=๐‘Ÿ.๏Š๏Šฑ๏Šง๏Š๏Šฑ๏Šง

Using the quotient rule for exponents, ๐‘ฅ=๐‘ฅ๐‘ฅ๏‰๏Šฑ๏Š๏‰๏Š, we can rewrite the right-hand side of our equation such that 1729=๐‘Ÿ๐‘Ÿ๏Š๏Šง

1729๐‘Ÿ=๐‘Ÿ.๏Š(3)

Next, we recall the formula for the sum of the first ๐‘› terms of a geometric series as ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๏Š๏Š. Let ๐‘†=1093๏Š; then, 1093=729(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ1093(1โˆ’๐‘Ÿ)=729(1โˆ’๐‘Ÿ).๏Š๏Š

From equation (3), we can substitute 1729๐‘Ÿ for ๐‘Ÿ๏Š: 1093(1โˆ’๐‘Ÿ)=729๏€ผ1โˆ’1729๐‘Ÿ๏ˆ1093โˆ’1093๐‘Ÿ=729โˆ’๐‘Ÿ364=1092๐‘Ÿ๐‘Ÿ=13.

The common ratio of the sequence is 13, which we can substitute back into equation (3) to calculate ๐‘›: 1729๏€ผ13๏ˆ=๏€ผ13๏ˆ๏€ผ13๏ˆ๏€ผ13๏ˆ=๏€ผ13๏ˆ.๏Š๏Šฌ๏Š

Using the product rule for exponents, ๐‘ฅ=๐‘ฅร—๐‘ฅ๏‰๏Šฐ๏Š๏‰๏Š, we can rewrite the left-hand side of our equation such that ๏€ผ13๏ˆ=๏€ผ13๏ˆ.๏Šญ๏Š

Therefore, ๐‘›=7.

The number of terms of a geometric sequence whose first term is 729, last term is 1, and sum of all terms is 1โ€Žโ€‰โ€Ž093 is 7.

In the fourth example, we need to find the geometric sequence, given information about some of its properties.

Example 4: Finding the Geometric Sequence given Its Last Term, Common Ratio, and the Sum of All Its Terms

Find the geometric sequence given the sum of all the terms is 3โ€Žโ€‰โ€Ž339, the last term is 1โ€Žโ€‰โ€Ž696, and the common ratio is 2.

Answer

We define the common ratio of a geometric sequence as ๐‘Ÿ. Let ๐‘Ÿ=2.

The last term ๐‘‡=1696๏Š, where ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง, and ๐‘‡ is the first term: 1696=๐‘‡(2).๏Š๏Šฑ๏Šง

Using the quotient rule of exponents, ๐‘ฅ=๐‘ฅ๐‘ฅ๏‰๏Šฑ๏Š๏‰๏Š, we can rewrite the right-hand side of our equation such that 1696=๐‘‡(2)(2)๏Š๏Šง

3392=๐‘‡(2).๏Š(4)

The sum of all the terms of the geometric series ๐‘†=3,339๏Š, where ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1๏Š๏Š: 3339=๐‘‡๏€น(2)โˆ’1๏…2โˆ’1๏Š

3339=๐‘‡(2)โˆ’๐‘‡.๏Š(5)

We know from equation (4) that ๐‘‡(2)=3392๏Š, so let us substitute this into equation (5) to create an equation in terms of ๐‘‡ that we can solve: 3339=3392โˆ’๐‘‡๐‘‡=53.

Therefore, the first term of the sequence is 53 and the common ratio is 2.

This means that the second term of the sequence is 53ร—2=106. The third term is 106ร—2=212. We find the following terms by multiplying the previous term by 2.

The geometric sequence is 53,106,212,โ€ฆ,1696.

In our final example, we are given a multiplicative relationship between two nonconsecutive terms in a geometric sequence, and the sum of two different nonconsecutive terms. We need to use our knowledge of geometric sequences in order to establish the first term and common ratio of the sequence and then find a finite sum of terms.

Example 5: Finding the Sum of n Terms of a Geometric Sequence under a Certain Condition

Find the sum of the first 7 terms of a geometric sequence given ๐‘‡=โˆ’8๐‘‡๏Šซ๏Šจ and ๐‘‡+๐‘‡=โˆ’64๏Šช๏Šฌ.

Answer

The ๐‘›th term of a geometric sequence, whose first term is ๐‘‡ and common ratio is ๐‘Ÿ, is denoted by ๐‘‡๏Š, where ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง.

This means that ๐‘‡=๐‘‡๐‘Ÿ๏Šซ๏Šช and ๐‘‡=๐‘‡๐‘Ÿ๏Šจ.

Substituting these expressions into the first equation, ๐‘‡=โˆ’8๐‘‡๏Šซ๏Šจ, we find ๐‘‡๐‘Ÿ=โˆ’8๐‘‡๐‘Ÿ.๏Šช

The first term of the geometric sequence is not equal to zero since the sum of two of its terms is nonzero. As ๐‘‡ and ๐‘Ÿ are nonzero, we can divide both sides of the equation by ๐‘‡๐‘Ÿ: ๐‘Ÿ=โˆ’8๐‘Ÿ=โˆ’2.๏Šฉ

Now that we know the value of the common ratio, we can use the second equation, ๐‘‡+๐‘‡=โˆ’64๏Šช๏Šฌ, where ๐‘‡=๐‘‡๐‘Ÿ๏Šช๏Šฉ and ๐‘‡=๐‘‡๐‘Ÿ๏Šฌ๏Šซ.

So, ๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ=โˆ’64.๏Šฉ๏Šซ

Substituting ๐‘Ÿ=โˆ’2, ๐‘‡(โˆ’2)+๐‘‡(โˆ’2)=โˆ’64โˆ’8๐‘‡+(โˆ’32๐‘‡)=โˆ’64โˆ’40๐‘‡=โˆ’64๐‘‡=85.๏Šฉ๏Šซ

Therefore, the first term of the sequence is 85 and the common ratio is โˆ’2.

We can now calculate the sum of the first seven terms of the geometric sequence using the formula ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ.๏Š๏Š

Let ๐‘‡=85 and ๐‘Ÿ=โˆ’2: ๐‘†=๏€บ1โˆ’(โˆ’2)๏†1โˆ’(โˆ’2)๐‘†=(129)3๐‘†=3445.๏Šญ๏Šฎ๏Šซ๏Šญ๏Šญ๏Šฎ๏Šซ๏Šญ

The sum of the first 7 terms of the geometric sequence is 3445.

We will finish this explainer by recapping some of the key points.

Key Points

  • A finite geometric sequence has the form ๐‘‡,๐‘‡๐‘Ÿ,๐‘‡๐‘Ÿ,โ€ฆ,๐‘‡๐‘Ÿ๏Šจ๏Š๏Šฑ๏Šง, where ๐‘‡ is the first term, ๐‘Ÿ is the common ratio, and ๐‘› is the number of terms in the sequence.
  • The ๐‘›th term of a geometric sequence is ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง.
  • The common ratio, ๐‘Ÿ, of a geometric sequence whose ๐‘›th term is ๐‘‡๏Š is given by ๐‘Ÿ=๐‘‡๐‘‡๏Š๏Šฐ๏Šง๏Š or ๐‘Ÿ=๐‘‡๐‘‡๏Š๏Š๏Šฑ๏Šง.
  • The sum of the terms in a sequence is called a series.
  • The sum of the first ๐‘› terms of a geometric sequence, with first term ๐‘‡ and common ratio ๐‘Ÿโ‰ 1, is denoted by ๐‘†๏Š, where ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š๏Š๏Šor

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