Derive a formula for the product of the first n members of an arithmetic progression with an initial element a and a common difference d;(a*d>0).
Verify the validity of your expression for (a,d,n)=(2,3,8).

Using the Gamma function, G(x) (G in the absence of a Greek gamma)....
Its property, G(x+1) = xG(x) gives, by induction,

G(a/d + n) = (a/d + n – 1)(a/d + n – 2)(a/d + n – 3)........(a/d)G(a/d)

so that the product of the first n terms of the AP will be:

a(a + d)(a + 2d)(a + 3d)......(a + [n – 1]d)  =      (d^n) G(a/d + n)/G(a/d)

I know this formula is not much help for evaluation purposes if you have no more than a basic calculator, but I’m going to cheat by using Maple for ‘validation’ purposes. So, for (a, d, n) = (2, 3, 8):

Product             = 3^8 G(2/3 + 8)/G(2/3)
            = 3^8 G(26/3)/G(2/3)    =  6561*19884.00733../1.354117939..
            = 96342400

...after all, is it really possible to devise a formula that allows you to evaluate this product, with only a basic calculator, more easily than multiplying the original terms together?  

Posted by Harry on 2012-02-22 22:04:17