A term in geometric series is formed by multiplying the previous term by a constant - which is called the **common ratio**.

For example, the series 2, 4, 8, 16, 32, ... has a common ratio of 2.

Because, .

In general,

### Examples

1. Find *x* if 5 + *x* + 45 + ... is a geometric series.

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For geometric series:

If *x* = 15, the series is 5 + 15 + 45 + ...

If *x* = -15, the series is 5 - 15 + 45 - ...

2. Is a geometric series?

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Therefore, the series is not geometric.

## Terms of a Geometric Series

### Examples

1. Find the 9th term of the series 2 + 6 + 18 + ...

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3. Which term of the series 4 + 12 + 36 + ... is equal to 78 732?

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[altex]a=4,\quad r=3\\ \\ { T }_{ n }=a{ r }^{ n-1 }\\ \\ 78732=4{ (3) }^{ n-1 }\\ \\ 19683={ 3 }^{ n-1 }\\ \\ \log _{ 10 }{ 19683 } =\log _{ 10 }{ { 3 }^{ n-1 } } \\ \\ \log _{ 10 }{ 19683 } =(n-1)\log _{ 10 }{ 3 } \\ \\ \cfrac { \log _{ 10 }{ 19683 } }{ \log _{ 10 }{ 3 } } =n-1\\ \\ 9=n-1\\ \\ n=10[/latex]

So the 10th term is 78 732.

4. The second term of a geometric series is 6 and the 5th term is 162. Find the first term and common ratio.

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## Partial Sum of a Geometric Series

The sum of the first *n* terms of a geometric series (*n*th partial sum) is given by the formula:

### Examples

1. Find the sum of the first 10 terms of the series 3 + 12 + 48 ...

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2. Evaluate .

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3. Evaluate .

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## Limiting Sum

In some geometric series, the sum becomes very large as *n* increases. This series **diverges**, or it has an infinite sum. However, some geometric series has a limiting sum, that is, there is a limit to the sum of all of the terms in the series up to infinity-th term. Yes, infinity! This series **converges** and has a limiting sum.

For example, the series , notice that after a while the terms are becoming closer and closer to zero and so will not add much to the sum of the whole series. Estimate its limiting sum.

For , as *n* increases, decreases and approaches zero e.g. when .

Using the example above,

### Example

Evaluate