Chapter 4

Question 1

infinite sequence [definition]

Answer 1

a list of intergers x_j for j ∈ N, denoted (x_j)^∞_j=m

Question 2

finite sequence [definition]

Answer 2

a list of numbers x_m, x_m+1, x_m+2,…,x_M-1, x_M ;
denoted (x_j)^M_j=m , where M ∈ Z with m ≤ M

Question 3

Answer 3

Question 4

Answer 4

(note: if m = n, this is interpreted as )

Question 5

definition of n! (“n factorial”) [definition]

Answer 5

Question 6

[finite series]

Answer 6

Question 7

[finite series]

Answer 7

Question 8

[finite series]

Answer 8

Question 9

[finite series]

Answer 9

Question 10

[finite series]

if a ∈ Z, then for all n ∈ N,

Answer 10

Question 11

[finite series]

Answer 11

Question 12

[finite series]

Answer 12

Question 13

[finite series]

Answer 13

Question 14

[finite series]

Answer 14

Question 15

binomial theorem [theorem]

Answer 15

Suppose k, m ∈ Z_≥0, with m ≤ k.
Then k! is divisible by m!(k-m)! .

Question 16

[theorem / formula]

Answer 16

“k choose m”

Question 17

binomial theorem for intergers [theorem / formula]

Answer 17

If, a, b ∈ Z and k ∈ Z_≥0, then

Question 18

[corollary] for k ∈ Z_≥0, we have

Answer 18

Question 19

Principle of mathematical induction—second form (“strong induction” [theorem]

Answer 19

For each k ∈ N, let P(k) be a statement. Assume that:

1. P(1) is true, and
2. if P(j) is true for all integers j such that 1 ≤ j ≤ n, then P(n + 1) is true.

Then P(k) is true for all k ∈ N.