Savings With Regular Investments using Geometric Sequence

1. Savings with Regular Investments Using 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.

EXAMPLE

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.

Geometric Sequence	Explanation
	The first term of the sequence, , is going to be the initial deposit of $1200. Year 1 deposit: $1200
	The second term of the sequence is going to be , or $1242. This represents the first year's deposit with 3.5% interest. We multiply the term by a growth factor of 1.035 to get this value. 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). Year 1: Year 2: $1200 Total: 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 gaining 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. So, 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. Year 1: Year 2: Year 3: $1200 Total: The second term now represents the $1200 deposit that was made one year after the initial deposit and which has gained 3.5% interest for only one year, for a value of $1242. 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 deposits, assuming no other deposits or withdrawals are made (and that the interest rate is fixed).

What is the value of the account after the 7th deposit, also keeping these assumptions? Or after the 11th deposit? Or even the 18th? We can use the following formula for the sum of a finite geometric sequence to answer this question.

formula to know

Sum of a Finite Geometric Sequence

EXAMPLE

We can use this equation to find the value of the account after seven deposits, or seven years, from the example above. Let's consider the known values:

• , the starting value of the account, is $1200.
• r, the common ratio, which in this context is the growth factor of the account (1 + annual percent rate of 3.5%), or 1.035.
• n, the number of years, is 7.

	Sum of a finite geometric sequence
	Substitute known values:
	Evaluate the exponent.
	Simplify the numerator and denominator.
	Evaluate the fraction.
	Multiply the values.

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

EXAMPLE

Suppose we start out by investing $1000 into an account. This account gains 4% interest each year, which is applied at the end of the year. At the beginning of each year, we add another $1000 to the account. As this pattern continues, we have $1000 being added to the account each year, and 4% interest applied to the balance of the account after each year. How much will be in the account at the end of 5 years?

In this example, we know the following values:

• , the starting value of the account, is $1000.
• r, the growth factor of the account (1 + annual percent rate of 4%) is 1.04.
• n, the number of years, is 5.

	Sum of a finite geometric sequence
	Substitute known values:
	Evaluate the exponent.
	Simplify the numerator and denominator.
	Evaluate the fraction.
	Multiply the values.

There will be $5,416.25 in the account after five years.

2. Real World Applications Using Infinite Geometric Sequence

Imagine a marble rolling across the floor. Measuring the distance the marble 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.

Suppose 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:

formula to know

Sum of a Infinite Geometric Sequence

Let's go back to the marble situation. We know the initial term, is 80, and the common ratio between each term, r, is one half, or 0.5.

	Sum of an infinite geometric sequence
	Substitute known values:
	Evaluate denominator.
	Evaluate fraction.
	Multiply values.

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

summary

In this lesson, you learned that periodically contributing to a savings or investment account is a great way to save money, for a variety of reasons. You learned how to model the growth of such an account by calculating savings with regular investments using a finite geometric sequence. You also explored real world applications using an infinite geometric sequence, such as modeling the rate of change of velocity of an object in motion. In the formulas for the sum of a finite geometric sequence and the sum of an infinite geometric sequence, is the first term in the sum, r is the common ratio between consecutive terms, and n is the number of terms.

Best of luck in your learning!