Exercises 65on all the money you owed during the month, and then your payment is subtracted.
So,An+, = An(1 + I) - P
Prove by induction thatAn, A - P)(I +/)n + -P
(b) Use this to calculate the monthly payment on a 30-year loan of $100,000 at 12%
interest per year. (Note that the formula is inexact, since money is always rounded
off to a whole number of cents. The derivation here does not do that. We use 12%
to make the arithmetic easier. You should consult a local bank to find a current
value.)- Sometimes, induction is not necessary for a proof, but an inductive proof can be sim-
pler than a noninductive proof. This is true for Examples 2 and 3 of Section 1.7.2.
(a) Find proofs of Examples^2 and^3 using familiar algebra but no explicit induction.
3(b) Optional: Find proofs of Examples 2 and 3 using calculus. (To some students
calculus may be more familiar than induction, but it is certainly more complicated
theoretically!)- Prove Theorem 4 of Section 1.5.4 in full generality. You may use Theorem 3 of Section
1.5.3, since it has already been proven. (Hint: Use induction on the number of sets). - For natural number exponents and nonzero bases, most of the familiar laws of expo-
nents can be proved by induction on the exponents using the facts that b° = I (for
b # 0) and bn+l = b .bn. Assuming that m and n are natural numbers and both r and
s are nonzero real numbers, prove the following:
(a) rm+n = rm rn
(b) rmn = (rm)n.
(c) If r > 1, then rm > rn if and only if m > n.
(d) If n,. r, s > 0, then rn > sn if and only if r > s.- A common use of induction is to prove various facts that seem to be fairly obvious but
are otherwise awkward or impossible to prove. These frequently involve expressions
with ellipses. Use induction to show that:
(a) X U (X 1 n X 2 fnX 3 n... n Xn) = (X U X 1 ) n (X U X 2 ) n... n (X U X,)
(b) X n (X 1 U X 2 U X 3 U-.- U Xn) = (X n X 1 ) U (X n X 2 ) U ... U (X n X,)
(c) (XI nX 2 n...nlXn) = X 1 U X2 U ... U Xn
(d) (X 1 U X2 U ... U X,) = X 1 n Y2 n ... n A n
38. (a) Prove that x E X0 n X 1 n ... n Xn if and only if x e Xi for every i such that
0<i <n.
(b) Prove that x E X0 U X 1 U ... U X, if and only if x E Xi for some i such that0< i <n.
(c) Use part (a) to give another proof of Exercise 37(a).3 We say explicit induction since, in the development of arithmetic from the foundations, almost everything about
- and • is proved by induction, including the familiar algebra needed for this problem.