Every now and then I put in the DVD for a lecture series on discrete mathematics and watch one. I like discrete math because a) it’s not calculus and b) it’s all about patterns, so it’s like doing a puzzle. I can actually wrap my head around discrete math. It suits the regimented side of me.
Anyway, today’s lecture started off about recognizing patterns in series of numbers added together. I knew (half my life ago) that there was a shortcut, but I never knew what it was or even thought about it after high school.
Professor Arthur Benjamin* showed an example series. Say you want to add up consecutive odd numbers starting with one. What’s a quick way to determine the sum without physically adding them all? You try to find the pattern.
1=1
1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
and so forth.
If you were to assign the number n to mean the number of numbers (sorry if that’s confusing) in the series, you’d start to see a pattern:
if n=1, then the series is 1=1, which is the same as 1=1²
if n=2, the series is 1+3=4, which is the same as 1+3=2²
if n=3, the series is 1+3+5=9, which is the same as 1+3+5=3²
and so on. So we figure out that if we add together n consecutive odd numbers starting with one, the sum will be n².
Let’s say we have the series:
1+3+5+7+9+11+13+15+17+19+21
We can count all the numbers, and we see there are 11. Eleven squared is 121, so the sum of that series is 121.
But what if you have a series that’s super long, and you don’t want to count how many numbers are in it? You find the pattern for the last number in the series and how it relates to n.
Again, the series:
for 1 , n = 1
for 1+3, n = 2
for 1+3+5, n = 3
for 1+3+5+7, n = 4
so,
when n = 1, the last number in the series is 1
when n = 2, the last number is 3
when n = 3, the last number is 5
and when n = 4, the last number is 7
If we puzzle it out, we’ll notice that by doubling n and subtracting 1, we’ll get that last number. So, for example, when n = 4, we get the last number, 7, by doubling 4 to make 8 and subtracting 1. The last number is 2n-1.
So now we can write the series this way:
1+3+5+7+…+(2n-1)=n²
I now have the shortcut. If I want to know what the sum of all odd numbers up to 99 are, I don’t have to write them out. I can say, “I know the last number is 99, so 2n-1 is 99. I can add one to make 100, divide it by 2 to get 50, and now I know n is 50. So the sum is 50 squared, which is 2500.”
I determined to figure out on my own the pattern for summing consecutive even numbers starting with 2, and in about 15 minutes of drawing it out, I did it! The answer is:
2+4+6+8+…+2n=n(n+1)
If I want to know the sum of all even numbers up to 100, I know that n=50, which makes the sum 50 times 51, or 2550.
I’ve never figured out a mathematical pattern by myself before, so I just wanted to share. It was kind of exciting!
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*Arthur Benjamin is a wonderfully enthusiastic teacher. You can check out his “Mathemagician” web page here.
Categories: Brain Workouts
Flabby Brain 