For many people, middle school math problems are a distant memory of a time when we stared at the piece of paper with our pencils in hand, desperately trying to find some sort of order in an apparently random sequence of numbers.
Your mission: find the answer to the problem below in less than 10 seconds—no calculator, just using logic. Ready, set, go!
Time’s up! How did you do? Did you manage to get it straight away? There are several ways to look at this problem, but we'll start with the most basic.
When we first look at the numbers, we notice that they don’t seem to have much in common. They aren’t prime numbers, they’re not divisible by each other, and they’re a mix of even and odd.
Ah, but if we look closer, there is a pattern. The first and third numbers are even, the second and fourth are odd. If we had to guess about the missing number, we would imagine it should be even.
Now, let’s look at the gaps between the numbers. To get 45, we would need to add 13 to 32. But if we add 13 + 45, we get 58, not 60 as we see in the sequence above. So what’s going on?
The difference between the next two terms is actually 15. So for terms 1 and 2, it’s 13, and terms 2 and 3, it’s 15. Now what about the last number in the sequence, 77?
How do we get there? Well, 77-60=17. But this doesn’t match up with the differences in previous numbers. Or does it?
The important aspect of sequences is not the numbers themselves but rather the pattern that they present. So far, we’ve got +13, +15, +17. See the pattern? The sequence seems to be constructed by adding the next odd number going up.
So if we follow this principle, we would need 77+19, which would give us 96, the correct answer!
But there’s an even cooler way to solve the problem that uses the logic of the sequence to do your work for you.
32+13=45. That much we got from the beginning. But look at the next part.
32+13+15=60. Pretty cool, huh?
32+13+15+17=77. Now we’re getting somewhere. And for the grand finale!
32+13+15+17+19=96. We’ve not only got the right answer but demonstrated how the sequence works in one line! Very nifty indeed.
Hope that you had fun working through this problem. But if you think that sequences are just a quick way to get math done in your day, you don’t have the whole picture.
In fact, simple problems like the following sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, helped unlock patterns in nature that gave birth to the Renaissance in Italy! This is the Fibonacci sequence, first developed in ancient India, then in Greece, and described in detail by Leonard Fibonacci of Pisa.
