Deriving reduction formula in Geometric Algebra

[Eric_H]

Hi, I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes.

I was reading the part about reduction formula (shown below) but couldn't get the result the shown in the book.
Can someone show me how iterating (1.15) gives the reduction formula?

notation in the book:
dot (.) is inner product
circumflex (^) is outer product

Thank you

 

[pat_connell]

.(1.15) A^B = A.B + A^B_a where A^B_a is the grade a part of A^B.The reduction formula in the book is:A^B_a = (1/2a) ∑ (−1)^b[A^B_b−A^B_{a+b}]where b goes from 0 to a−1.