# What is a geometric sequence? Definition and examples

What is a geometric sequence? A geometric sequence is an ordered list of non-zero numbers in which the first number is picked or known and then each number afterward is found by multiplying the previous number by the same non-zero constant or fixed number.

For example, the ordered list of numbers shown below is a geometric sequence.

2, 10, 50, 250, 1250, ...

The first number is 2 and it is a number that I picked.

The second number is found by multiplying 2 by 5 to get 10

The third number is found by multiplying the second number or 10 by 5 to get 50.

And so forth...

A geometric sequence is also called geometric progression.

## More examples showing what a geometric sequence is

**a.**

2, 6, 18, 54, 162, ...

The first number is 2 and then keep multiplying each previous number by 3 to get the next number.

**b.**

48, 12, 3, 0.75, ...

The first number is 48 and then keep multiplying each previous number by 0.25 or 1 / 4 to get the next number.

**c.**

3, -6, 12, -24, 48

The first number is 3 and then keep multiplying each previous number by -2 to get the next number.

## What is a geometric sequence in real life?

A man invests some money in the stock market. His initial investment is 5000 dollars. Suppose the amount at the beginning of the second year is 150% of the initial investment. Then, each amount is 150% of the previous amount. How much money does he have after 4 years?

**Solution**

The first number of the geometric sequence is 5000. Then, each number after that is multiplied by 150% or 150 / 100 = 1.50.

5000, 7500, 11250, 16875, 25312.5, ...

After 4 years, he will have 25312.5 dollars.

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