Up and Down Series
David W. Hansen
© 2003
The following set of beautiful equations are called Up and Down Series because of the up and down
nature of the terms in each series on the left.
1 = 1 = 12
1 + 2 + 1 = 4 = 22
1 + 2 + 3 + 2 + 1 = 9 = 32
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 42
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 = 52
Here’s a dot picture of the 4th up and down series (which looks like a tree!)
.
. . .
. . . . .
. . . . . . .
1 + 2 + 3 + 4 + 3 + 2 + 1
The five equations above suggest to us that, in general,
1 + 2 + 3 + . . . + n + . . . + 3 + 2 + 1 = n2, for any natural number n.
Let’s try to prove this.
Since 1 + 2 + 3 + . . . + n = n(n+1)/2, then we get
1 + 2 + 3 + . . . + n + . . . + 3 + 2 + 1
= 1 + 2 + 3 + . . . + n + n + . . . + 3 + 2 + 1 – n
= (1 + 2 + 3 + . . . + n) + (n + . . . + 3 + 2 + 1) – n
= n(n+1)/2 + n(n+1)/2 – n
= 2[n(n+1)/2] – n
= n(n+1) – n
= n2 + n – n
= n2
which proves our conjecture.
Example 1. Find an up and down series whose sum is 64. The sum of an up and down series is n2,
so n2 = 64, and n (the middle or largest term) = 8. The series is then
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 82 = 64.
Example 2. Find the number of terms in an up and down series whose sum is 256. Since n2 = 256
= 162, the middle number of the series is n = 16. There will then be15 terms to the left and 15 terms
to the right of this middle number, giving us a total of 31 terms (including, of course, the middle number).