Solution:

445

2=(1+1)1
6=(2+1)2
21=(6+1)3
88=(21+1)4

So 445=(88+1)5

Solution:

100

list of consecutive numbers that does not contain a 'T' in the spelling

Solution:

192 021

If you look at it as two-digit numbers you'll see:10 11 12 13 14 15 16 17 18 19 20 21

Solution:

South

The series: south, east, north, south, west, east spirals clockwise from the top left-hand corner.

Solution:

Next in sequence is 13112221.

This series describes each number that came before it:
1 [One]
11 [One One]
21 [Two One(s)]
1211 [One Two and One One(s)]
111221 [One One, One Two, and Two One(s)]
312211 [Three One(s), Two Two(s), and One One]

The next number in the series is:
13112221 [One Three, One One, Two Two(s), Two One(s)]

Solution:

N for November

M: March
A: April
M: May ....and so on ...

Solution:

794

The nth term in the series is given by the sum of x^n for x in the range 1 to 3, i.e. 1^n + 2^n + 3^n.

Thus the first term is 1^1 + 2^1 + 3^1 = 1 + 2 + 3 = 6,
the second term is 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14,
and so on.

The sixth term is then 1^6 + 2^6 + 3^6 = 1 + 64 + 729 = 794.

Solution:

A and W.

The letters are the first letters of the words in the question. Thus, the next two letters are A and W.

Solution:

61

Reversed digits of squares of number in descending order.

Solution:

129.

These are the first 10 prime numbers (2, 3, 5...) prefixed with a 1