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Sum of a Geometric Sequence in the Real World

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This lesson applies formulas for the sum of geometric sequences in real world scenarios.

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- Sum of a Finite Geometric Sequence
- Sum of an Infinite Geometric Sequence

**Sum of a Finite Geometric Sequence**

Periodically contributing to a savings or investment account is a great way to increase your wealth, prepare for unexpected emergencies, or plan for the future. We may be able to model the growth of such an account using a geometric sequence.

Suppose we start out by investing $1200 into an account. This account gains 3.5% interest each year, which is applied at the end of the year. At the beginning of each year, we add another $1200 to the account. As this pattern continues, we have $1200 being added to the account each year, and 3.5% interest applied to the balance of the account after each year.

How can we model this using a geometric sequence?

The first term of the sequence is going to be the initial deposit of $1200. The second term of the sequence is going to be 1200 • 1.035, or $1242. This represents the first year's deposit with 3.5% interest. With two terms in the sequence now, the first term actually has a different meaning. It now represents the $1200 that is added after Year 1 (and it hasn't gained any interest yet, because it has just been deposited).

If we add the two terms together, we have a value of $2442, which represents two deposits of $1200, one of which has been in the account for a year, thus has gained 3.5% interest.

Let's think about the third term in the sequence. We take the second term, $1242, and multiply it once again by 1.035 to show its growth. The third term has a value of $1285.47. This represents the initial deposit (now made 2 years ago) that has gained 3.5% interest for two years.

The second term now represents the $1200 deposit that was made one year after the initial deposit, and has gained 3.5% interest for only one year. The first term always represents the most recent deposit of $1200, having gained no interest.

As we can see, when we add the terms together, we are finding the value of the account after n number of years, assuming no other deposits or withdrawals are made (and that the interest rate is fixed).

What is the value of the account after 7 years, also keeping these assumptions? We can use the formula for the sum of a finite geometric sequence to answer this question. Our formula is:

where,

- is the sum of n terms (in this context, the balance of the account after n years)
- is the value of the first term in the sequence (in this context, the starting value of the account)
- is the common ratio (in this context, it is the growth factor, 1 + the annual percent rate, expressed as a decimal).
- is the number of terms (in this context, it is the number of years)

This means that after 7 years, the account will have a balance of $9,335.29

**Sum of an Infinite Geometric Sequence**

Imagine a marble rolling across the floor. Measuring the distance the marvel travels in constant intervals, we notice that the marble travels half of the distance traveled in the previous interval. As the marble continues to roll, it will travel shorter and shorter distances within these time intervals, and will eventually travel a distance of virtually zero. This means that the total distance traveled (the sum of all of the recorded distances) will converge to a specific distance.

The following sequence describes the distances traveled during each time interval, measured in centimeters:

How far does the marble travel in total? To answer this question, we will find the sum of this infinite geometric sequence, using the following formula:

The initial term, a_{1} is 80, and the common ratio between each term, r, is one half, or 0.5.

This means that the total distance traveled by the marble will eventually converge to 160 centimeters.