30!

PART 1.3: Advanced Mathematical Expansion Rates
POSTULATE 3.1: the “Sequential set of Mulitplicatives”
counts every other multiplicative factor in sequence.
AXIOM 3.2: the sequence begins with the line 1 X 2 = 2,
then skips one place ahead to the line 2 X 2 = 4, etc.
LEMMA 3.3: here the sequence is depicted extending to
limit-15, on the line 30 = 2 X 15, in the lower-right.
THEOREM 3.4: the sequential set is 1/2 the sums on the
opposite side, is base-2, and comprises the “Even set.”
POSTULATE 3.5: the “Odd set of Multiplicatives”
counts every other multiplicative factor as an Odd #.
AXIOM 3.6: the sequence begins with the first line,
1 X 1 = 1, the skips one place to the line 3 X 1 = 3.
LEMMA 3.7: here the sequence is depicted extending to
limit-31, the last interval in the lowest-right chart.
THEOREM 3.8: the Odd set is 1 X the same # opposite it
in the Sequential set, is base-1, & compliments “Evens.”
POSTULATE 3.9: the “Sequential Set of Additives” are
the #’s inner-most on the chart, counting in sequence.
AXIOM 3.10: the sequence begins with 1, next 2, etc.
LEMMA 3.11: the sequence here ends with limit-30.
THEOREM 3.12: the products of the multiplicative
factors are herein interpolated, both Even and Odd.
POSTULATE 3.13: the “Factorial Set of Additives” is =
to the Even set of Multiplicatives X (times) the Odd set.
AXIOM 3.14: the sequence begins with 1 X 1 = 1, but is
not differentiated until line 1 X 3 = 3, skip to 3 X 2 = 6.
LEMMA 3.15: here the sequence ends with limit-465,
the last sum and factor in the lower right of the chart.
THEOREM 3.16: the Factorial Set is a ratio of the Even
and Odd multiplicative sets, and thus combines both.
POSTULATE 3.17: the complete chart given here graphs
arithmetic and exponential expansion rates both.
AXIOM 3.18: the Even Multiplicatives yield arithmetic
expansion, & the Odd Multiplicatives yield exponential.
LEMMA 3.19: the Even set is base-2, and the Odd-set is
base-1; the Factorial Additive set is sums between them.
THEOREM 3.20: for any # n; if n = Even, then n + (n -1)
+ (n - 2) + (n - 3), ... [n - (n - 1) = 1], etc. = n X (2n + 1).
THEOREM 3.21: for any # n; if n = Odd, then n + (n -1)
+ (n - 2) + (n - 3), ... [n - (n - 1) = 1], etc. = 2 (n + 1) + 2
(n - 1) = 4n.