These are the successive

triangular numbers formed from the sequence:

The three lessons are based on investigations into

triangular numbers, providing an alternative to the linear functions work we generally see in this area.

The author covers Pythagorean triples, the Monty Hall problem and other probability theory puzzles, the Lucas sequence, the

triangular numbers, the square numbers, and many other related subjects.

We also give an application of the proposed fuzzy ordering method to the fuzzy risk analysis problem, where the evaluating values are represented by

triangular numbers.

In this method according to the number of options the membership function is considered as the following illustration and

triangular numbers are also determined in this way.

The squares make up the perfect pangram I'D PAY, but despite a good consonant-vowel balance it is difficult to get anything out of the

triangular numbers, for which the relevant letters are ACFJOU.

Similarly to right and left

triangular numbers we can introduce right and left trapezoidal numbers as parts of a trapezoidal number.

A mate went for the same effect, but his odd

triangular numbers looked like bushy wedges of Dairylea.

Well done to Dominic Degnan from County Durham who correctly worked out that 1, 3, 6, 10, 15, 21 are

triangular numbers in last fortnight's quiz.

The second vector is the sum total of

triangular numbers of the above matrix as inversed.

The

triangular numbers are formed by partial sum of the series 1 + 2 + 3 + 4 + 5 + 6 + 7.

Torres and Viorica Teca [1] have further investigated these sequences and defined mirror and symmetric Smarandache sequences of

Triangular numbers making use of Maple system.