which is known as Euler's pentagonal number theorem, and also (3) and

g-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue's identity and Euler's pentagonal number theorem.

It is well known that a positive integer N is called a Pentagonal Number if N = m(3m-1)/2 for any integer m > 0, and a positive integer N is called a Generalized Pentagonal Number if N = m(3m-1)/2 for any integer m.

5.1 Theorem: (i) [Q.sub.n.sup.(1)] is Generalized Pentagonal Number if and only if n [equivalent to] 0, 1 or 3 (ii) [Q.sub.n.sup.(1)] is Pentagonal Number only for n = 0 or 1.

Some third of the material is concerned with biographical and other contextual issues, while the bulk of the selections focus on particular aspects of Euler's contributions to mathematics, including infinite series, the zeta functions, Euler's constant, differentials, multiple integrals, the calculus of variations, the

pentagonal number theorem, quadratic reciprocity, and the fundamental theorem of algebra.

The n-th pentagonal number [t.sub.n]; n [member of] N of dimension 2 is defined by [t.sub.n] = 2n + 5/2 2n(n-1)-[n.sup.2].

Working on the same lines we have defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of pentagonal numbers of dimension 2 using the Maple system .

So, how do you get from one

pentagonal number to the next one?

Now your students have the background to try the next step, which involves the

pentagonal numbers (N = 5).

Again it has been found that the

pentagonal numbers n(3n-1)/2 for n [greater than or equal to] 1 have some structures like a graph but it does not mean a graph as it does not satisfy the basic property of graph theory, that is, sum of the degrees of a graph is equal to the twice of edges.

One of the authors A.S.Muktibodh [2] working on the same lines has defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of

pentagonal numbers of dimension 2 using the Maple system.

Following directly on from the triangular and square numbers are the

pentagonal numbers P(n), and these will now be developed geometrically.