1.3 Natural Numbers 2712. Finite Geometric Sums: If r-:/:-1, a+ ar + ar^2 + ar^3 + · · · + arn =
a - arn+l
1-r
13. 2n :S (n + 1)!.
Bernoulli's Inequality: for any fixed x > -1, (1 + x)n :'.;:: 1 + nx.
For any fixed x :'.;:: 0, (1+x)n:'.;::1 + nx + ~n(n - l)x^2.
13n - 5n is divisible by 7.
2^2 n-l + 1 is divisible by 3.
18. Prove Theorem 1.3.11.- xn - yn = (x - y)(xn-1 + xn-2y + xn-3y2 + ... + xyn-2 + yn-1 ). [Hint:
start with n = 2 and use Theorem 1.3.11.] - Vm, n E N, a man = am+n. [Hint: let m be a fixed natural number, and
use mathematical induction on n; see Definition 1.3.12.] - Vm, n EN, (am)n = amn. [See hint for Exercise 20.]
- 'in EN, anbn = (ab)n.
- Binomial Coefficients:
(a) Factorials: 'in E N, we use mathematical induction to define n
factorial (denoted n!): l! = 1, and (n + 1)! = n!(n + 1). We also
define O! = l. Show that 'in EN, n! = 1 · 2 · 3 · ... · n.
(b) \;/ k :S n in N, we define the binomial coefficient (~) by the formula
(~) = k!(nn~ k)!. Since O! makes sense, we define(~) by letting k =
0 in this definition. Verify the familiar formulas, (~) = 1, G) = n,
(~) = (n'.:'..k), and (k'.:'.. 1 ) + C) = (nkl). Relate these identities to the
famous "Pascal's triangle."
( c) Prove that under this definition, (~) is always a natural number.
[Use mathematical induction on n and the last identity in (b).]
n- The Binomial Theorem: Recall the summation notation, I: ak
k=l
a 1 + a 2 +···+an, and its obvious extensions to sums starting with k = 0
or k = any other natural number. Use the results of Exercise 23 to prove
the binomial theorem,