Consider the recurrence relation

$.$ Assume that

is a power of

$,$ that is$,$

for some integer

$.$

$($a$)$

If the levels of an expansion of the recurrence relation are numbered

through

$,$ then what is the value of

$?$ Express your answer in terms of

$.$

$($b$)$

If the levels of an expansion of the recurrence relation are numbered

through

$,$ then how many terms are there at level

$?$