Learn what recurring investments are, how to use the compound interest formula to calculate how much money they make, and how you can calculate how fast your money will grow.
If you’re under the impression that mathematical series like those we talked about last week aren’t useful in the real world or that the compound interest formula we talked about a few weeks ago is just a one trick pony, the next few articles should change your mind. Though mathematical series are certainly a beautiful and elegant part of math, we’ll see that they’re also incredibly useful. And although we already know the compound interest formula can be used to calculate simple interest, we’re going to find out that it can also be used to do much more.
How to Calculate How Fast Your Money Will Grow
In fact, as we’ll see over the next two articles, if we put these two ideas together we gain even greater abilities. In particular, we can use the combined power of the compound interest formula and mathematical series to calculate how fast your money will grow in something called a recurring investment—and that turns out to be a very useful real world skill!
Recap: How to Calculate Simple Interest
As we’ve seen in previous articles, the compound interest formula can be used to calculate what’s called the simple interest earned in investments like a bank savings account. To refresh your memory, the compound interest formula says that
FV = PV x (1 + rate)^years
So if you invest $2000 at the beginning of the year in an account earning 5% annual interest, the compound interest formula says that at the end of each of the first 3 years, the account will be worth:

Value at the End of Year 1 = $2000 x 1.05^1 = $2100.00

Value at the End of Year 2 = $2000 x 1.05^2 = $2205.00

Value at the End of Year 3 = $2000 x 1.05^3 = $2315.25
What Is a Recurring Investment?
Notice that in the investment we’ve been talking about, money was only put into the account once at the beginning of the first year. After that, the account grew only by accruing earned interest. But that’s not how most people actually invest their money. Instead, people usually put some fraction of their income into an investment account every year.
For example, if you put $2000 into an account at the beginning of every year, not only will that account grow by earning interest, but it will also grow simply due to the fact that you’re putting in more and more money. This type of investment where money is periodically put into an account is called a recurring investment. And, as we’ll see in a minute, it’s a great way to grow your money.
How to Calculate the Value of a Recurring Investment
To see why a recurring investment is such an effective means of growing your money, let’s think about how to calculate the value of such an investment after some number of years. In particular, let’s say that you invest $2000 in an account earning 5% interest…exactly as you did before. But this time, let’s also say that you put an additional $2000 into the account at the beginning of each subsequent year. Using the compound interest formula, we find that the value of the account at the end of the first 3 years is:

Value at the End of Year 1 = $2000 x 1.05^1 = $2100

Value at the End of Year 2 = $2000 x 1.05^1 + $2000 x 1.05^2 = $4305

Value at the End of Year 3 = $2000 x 1.05^1 + $2000 x 1.05^2 + $2000 x 1.05^3 = $6620.25
So, at the end of 3 years you’ve invested $6000 of your money into the account, which means that you’ve earned $6620.25 – $6000.00 = $620.25 of interest. When you compare that to the $315.25 of interest that you earned in the account where you only invested one lump sum amount at the outset, you begin to see the real power of recurring investments. Not only do you earn more interest, but you also save more of your money to begin with!
Is There an Easier Way to Calculate Recurring Investment Values?
Okay, it’s nice that we figured out how to calculate the value of a recurring investment after 3 years, but how would you go about figuring out the value of such an investment after something like 20 years? Figuring that out by hand using the method we used for 3 years would be a huge pain. And that naturally leads to the question: Is there a better way? Perhaps something like the compound interest formula…but for recurring investments? Well, you’ll be happy to know that there is a better way—but you’ll have to wait until next week to find out what it is, and what mathematical series have to do with it.
Number of the Week
Before we finish up for today, it’s time for this week’s featured number selected from the various numbers of the day that I posted to the Math Dude’s Facebook page over the past week. This week’s number is 93%. What is that? Well, somewhat sadly, it’s the approximate percentage of all sent email messages that are spam! Believe it or not, there are approximately 300 billion email messages sent every day. Which means that about 0.93 * 300 billion = 279 billion spam email messages are sent every day. I don’t know about you, but that number really makes me appreciate spam filters!
New FREE Math Dude eBook!
But enough about spam, I’ve got a very exciting announcement that is most definitely not spam. As you may have heard, I have a new book coming out next month called the Math Dude’s Quick and Dirty Guide to Algebra. It’s available right now for preorder from Amazon, Barnes & Noble, and all the usual places.
But, believe it or not, that’s NOT today’s exciting announcement. No, today’s announcement is that I’ve also written a free ebook called Why Math Isn’t an Awful Nerd. It’s available to download for FREE right now from Amazon. What’s it about? Well, the title really says it all: A lot of people have had bad experiences with math, and they feel like it’s just a painful and awful subject. I understand where they’re coming from, but I also know that it really doesn’t have to be that way.
So, using lots of visual explanations, puzzles, and games, this new ebook explains stepbystep exactly how math works, why it’s so awesome, and hopefully it’ll convince all those people thinking that math is awful to change their minds and agree with me that math is—dare I say it—elegant and even beautiful. Be sure to download your copy of Why Math Isn’t an Awful Nerd today...while it’s still free. Thanks for checking it out!
Wrap Up
Okay, that’s all for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new number of the day and math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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