Proof Without Words: Sum of Triangular Numbers

36
MATHEMATICS MAGAZINE
Proof Without Words: Sum of Triangular
Numbers
Á N G E L P L A Z A
Universidad de Las Palmas de Gran Canaria
Las Palmas, Canarias, Spain
[email protected]
Theorem.
n
n+1
n(n + 1)
n(n + 1)(n + 2)
=
.
⇒
Tk =
2
2
6
k=1
Tn = 1 + 2 + · · · n =
Proof.
n
Tk
∑
k =1
Tn 1
2
3
T3 1
T2 1
T1 1
2
2
3
n−1
1
2
1× n
2×( n −1)
1× n
n ×1
1
n−1
...
2×(n−1)
n+2
1×n
k =1
(n−1) × 2
2
n
3∑ Tk
n
n ×1
2× (n−1)
1× n
...
...
(n−1) × 2
n ×1
n ×1
2×( n −1)
n ( n + 1)
2
c Mathematical Association of America
Math. Mag. 89 (2016) 36–37. doi:10.4169/math.mag.89.1.36. MSC: Primary 11Y55.
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37
VOL. 89, NO. 1, FEBRUARY 2016
This proof is close to, and it can be seen as a variation of, Zerger’s proof [1], which
also appears on page 94 of Nelsen’s compendium of PWWs [2].
REFERENCES
1. M. J. Zerger, Proof Without Words: Sum of Triangular Numbers, Math. Mag. 63 no. 5 (1990) 314.
2. R. B. Nelsen, Proof without Words: Exercises in Visual Thinking. The Mathematical Association of America,
Washington, DC, 1993.
Summary. The
triangular numbers are given by the following explicit formulas: Tn = 1 + 2 + · · · n =
n(n+1)
= n+1
2 . Here it is proved visually that
2
n
k=1
Tk =
n(n + 1)(n + 2)
.
6
DR. ANGEL PLAZA (MR Author ID: 350023) received his masters degree from Universidad Complutense
de Madrid in 1984 and his Ph.D. from Universidad de Las Palmas de Gran Canaria in 1993, where he is a
Full Professor in Applied Mathematics. He is interested in mesh generation and refinement, combinatorics and
visualization support in teaching and learning mathematics.
From the Files of Past MAGAZINE Editors
J. Arthur Seebach and Lynn Arthur Steen 1976–1980
As chairman of the MAA’s Publications Committee, Ed Beckenbach asked then
M AGAZINE co-editors Lynn Arthur Steen (LAS) and J. Arthur Seebach, Jr. (AS)
what ideas they had for improving the M AGAZINE. They wanted to make it more
public-oriented, but they realized they had no writers and no audience for such a
magazine. Instead, they made some cosmetic changes, like putting something other
than the table of contents on the cover, itself a controversial decision.
LAS and AS wanted to have articles start at the top of a page instead of simply
starting where the previous article ended; ideally they would start at the top of a
right-hand page. They needed some short things (called filler) to insert to fill space
at the ends of some articles. At a suggestion of Roger Nelsen, they began including
some Proofs without Words as filler. Proofs without Words have been a mainstay of
the M AGAZINE ever since.
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