Fascinating Triangular Numbers

 

Displaying Triangular Numbers

The numbers which can be arranged in a compact triangular pattern are termed as triangular numbers. The triangular numbers are formed by partial sum of the series 1+2+3+4+5+6+7......+n. So

T1 = 1
T2 = 1 + 2 = 3
T3 = 1 + 2 + 3 = 6
T4 = 1 + 2 + 3 + 4 = 10

So the nth triangular number can be obtained as Tn = n*(n+1)/2, where n is any natural number.In other words triangular numbers form the series 1,3,6,10,15,21,28.....

Flocks of birds often fly in this triangular formation. Even several airplanes when flying together constitute this formation. The properties of such numbers were first studied by ancient Greek mathematicians, particularly the Pythagoreans.

Have you heard of the following famous story about the famous mathematician Carl F. Gauss.

" The teacher asked everyone in the class to find the sum of all the numbers from 1 to 100. To everybody's surprise, Gauss stood up with the answer 5050 immediately. The teacher asked him as to how it was done. Gauss explained that instead of adding all the numbers from 1 to 100, add first and last term i.e. 1 + 100 =101, then add second and second last term i.e. 2 + 99 =101 and so on. Every pair sum is 101 and their will be 50 such pairs ( total 100 numbers in all to be added), so 101 * 50 = 5050 is the answer. So the sum of numbers from 1 to N is (N/2)*(N+1), where N/2 are the number of pairs and N+1 is sum of each pair. This the famous formula for nth triangular number."

Some of the interesting properties of triangular numbers published in [5] are:

Curious properties of Triangular Numbers:

T1 + T2 = 1 + 3 = 4 = 22
T2 + T3 = 3 + 6 = 9 = 32

9*T1 + 1 = 9 * 1 + 1 = 10 = T4
  9*T2 + 1 = 9 * 3 + 1 = 28 = T7

8*T1 + 1 = 8 * 1 + 1 = 9 = 32
  8*T2 + 1 = 8 * 3 + 1 = 25 = 52

T42 = 102 = 100 = 13 + 23 + 33 + 43
T52 = 152 = 225 = 13 + 23 + 33 + 43 + 53

21 + 15 = 36 = T8 : 21 - 15 = 6 = T3

171 + 105 = 276 = T23 : 171 - 105 = 66 = T11

703 + 378 = 1081 = T46 : 703 - 378 = 325 = T25

     and so on.

T55 = T10 + T54 = T1540 - T1539 = T7 * T10

T75 = T29 + T69 = T77 - T17 = T5 * T19

     and so on.

 

Some New Observations on Triangular Numbers :

 

If you find any new and interesting observation about triangular numbers, please email me.

---------------------------------------------------------------------------

References:

[1] Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.

[2] Beiler, Albert H. Recreations in the Theory of Numbers. New York: Dover, 1966.

[3] Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.

[4] Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

[5] Gupta, Shyam Sunder "Curious Properties of triangular numbers." Science Reporter , September 1987, India.

[6] Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138 and 147-150, 1994.

[7] Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

[8] Madachy, Joseph S. Madachy's Mathematical Recreations. New York: Dover, 1979.

[9] Pickover, Clifford A. Wonders of Numbers. New York: Oxford University Press, 2001.

[10] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

[11] Trotter, T. Jr. "Some Identities for the Triangular Numbers." J. Recr. Math. 6, 128-135, 1973.

[12] Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1997.

[13] Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin Books, 1991.



In Association with Amazon.com

This page is created on 26 Oct, 2002.

Back Home