15. With Pn defined as in the chapter by Pn1 Py j1 j n1 rj :...

15. With Pn defined as in the chapter by Pn1 Py j¼1 j nð1 þ rÞj :

(a) Derive (6.30), Pn ¼

1 r

Xn1 i¼0 n

i Pi þ 1

" #

:

(Hint: Note that Xy j¼1 j nð1 þ rÞj ¼ ð1 þ rÞ1 þ ð1 þ rÞ1 Xy j¼1 ð j þ 1Þnð1 þ rÞj and expand ð j þ 1Þn with the binimial theorem of (7.15) as seen in this chapter’s derivation.)

(b) Develop explicit formulas for Pn, n ¼ 2; 3; 4; 5 using P0 ¼ 1 r and P1 ¼ 1þr r2 .