# A geometric sequence as a model

A geometric sequence is a number sequence in which the ratio between two consecutive terms is constant. Each term is found by multiplying the previous term with a number called the common ratio.

# 2,4,8,16,32,...

Above is a geometric sequence, and the common ratio of two consecutive terms is 2. This number is marked by q. The sequence would continue 64,128,256,512, ...

Example 1

The first term of a geometric sequence is 2 and the second term is 6. Find the next two terms of the sequence. ### General term

In a geometric sequence, we have a common ratio q that multiplies the term of the sequence to get the next term. To get the second term we multiply the first term by q. To get the third term we multiply the first term twice by q. The fourth term is obtained by multiplying the first term three times by q . And so on.

We can get any term when we multiply the first term by q to the power less than one of the order number of that term. ## The sum of a geometric series

The sum of a geometric series can be calculated by the formula below.

We need to know the first term, the common ratio between the terms and the number of terms to be added together. ## A Geometric sequence as a model

Example 2

€100 is deposited into the account at the beginning of each month for a year. The monthly interest rate on the account is 0,5%. How much money is in the account at the end of the year? The first deposit will be made in early January.

Let’s look at each deposit separately. The deposit becomes 1,005 times greater a month. The first deposit increases by the interest rate for 12 months and the last deposit only for one month. The monthly values of the deposits form a geometric series. The last deposit, in December, is marked as the first term. The ratio is 1,005 and the number of terms is 12. Now we calculate the sum of the geometric series There is €1239,72 at the end of the year

Example 3

The rent for an apartment is €800. The rent is increased 5% annually. How much money is required for rent over 10 years?

The rent will be 1,05 times greater a year. Money needed for rent every year The amount of money needed for rent each year forms a geometric series. The common ratio is 1,05 and the first term is 9600. The number of terms is 10.

The sum is calculated by so, in 10 years, the money needed for rent is €120 747,77.

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