Proof Without Words: Sums of Every Third Triangular Number Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR Let Tk denote the kth triangular number, Tk = 1 + 2 + · · · + k. We show that T3 + T6 + · · · + T3n = 3(n + 1)Tn . Proof. I. T3k = 3(k 2 + Tk ).

Tk

3k

k2

Tk

k2

II.

n 

k2

Tk

(k 2 + Tk ) = (n + 1)Tn .

k=1

Tn ... T3 T2 T1 1 1

9

4 2

Exercise: Show that

3 n 

n2

... ...

T3k−1 = 3nTn and

k=1

n+1

n n 

T3k−2 = 3(n − 1)Tn + n.

k=1

Summary. A visual proof relating sums of every third triangular number to a single triangular number. http://dx.doi.org/10.4169/college.math.j.46.2.98 MSC: 11B65

98

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Proof Without Words: Sums of Every Third Triangular Number Roger B ...

Let Tk denote the kth triangular number, Tk = 1 + 2 +···+ k. We show that. T3 + T6 +···+ T3n = 3(n + 1)Tn. Proof. I. T3k = 3(k2 + Tk). k. 2. T k. 3k k. 2 k. 2. T k. T k. II. n.
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