Comments on: Triangular Numbers and the Devil

By: Brian

Brian — Fri, 12 Mar 2010 14:17:53 +0000

Hey, Thanks for the excellent explanation. I appreciate it, and I’m sure other readers will, too.

By: sc

sc — Fri, 12 Mar 2010 14:01:17 +0000

for series 1,2,3, ….. ,n

1) sum of series = (n*(n+1))/2
eg from 1 to 19,
sum = (19*20)/2 = 190

The “diagram” below shows the reasoning behind this for the series 1,2,3,4. The ‘*’ represents the value of numbers 1,2,3 and 4. To find the number of *s, add an equal number of ’0′s to form a rectangle. This will form a 4 x 5 rectangle. The number of *s is given by (4*5)/2

12345
1 *0000
2 **000
3 ***00
4 ****0

2) sum = ((1+n)/2) * n
sum = ( (1+19)/2 ) * 19 = 190

This finds the average of the series (1+n)/2 and multiply the average by the number of terms in the series. Average of 1 to 19 is (1+19)/2 or 10. Multiply by 19 to get the sum of 190.

3) Multiply the median number of the series by the number of terms in the series. This is somewhat similar to (2) above.

If n is odd, the median number is (n+1)/2. The median number for 1 to 19 is (19+1)/2 or 10. Sum of 1 to 19 is 10 * 19 = 190.

If n is even (eg 1 to 20), there is no exact median number. Therefore to get the sum of 1 to 20, first get the sum of 1 to 19 (as above) then add the last number 20 to get the final sum. sum of 1 to 20 is 190 + 20 = 210

By: Brian

Brian — Fri, 12 Mar 2010 12:22:24 +0000

Peter, Right on the money. This is also the way that is described on this Mathmojo page about triangular numbers.

By: Peter Ash

Peter Ash — Fri, 12 Mar 2010 03:48:54 +0000

The standard way to do this (seen in many high school math texts) is to write all the numbers in the sum in increasing order, and under that write down all the numbers in the sum in decreasing order. If n = 20, as in the problem given, each column adds to 21 and there are 20 columns, so the total is 420. Since we wrote the sum down twice, the sum of the of the numbers is 420/2 = 210.

I tell my students this is just like counting the number of people in a crowd by counting the total number of legs and dividing by two.

By: Mia

Mia — Wed, 03 Mar 2010 01:34:57 +0000

I read our library’s copy of The Number Devil a couple of years ago with our oldest daughter – it was fun and interesting! Thank you for bringing up this subject.