64
CHAPTER 1 Sets, Proof Templates, and Induction
(c) Fo + F 3 +
+ F 3 = F 3 n+ 2 /2 for n > 0
(d)F 1 =Fn "
- (-1)n forn >_ 0
25. The Lucas numbers are defined as LO = 2, L, = 1, and Ln = Ln-1 + Ln-2 for n >
2. Prove the following identities for Lucas numbers.
(a) Lj + L 2 + .. • + Ln = Ln+ 2 - 3 for n > 1
(b) L2
2
2 ...
2-forn>2
(c) L 2 + L 4 + - "- +L 2 n = L 2 n+l - 1 for n > 2
26. Find the value of the following sums:
(a) 2 + 2 -
3n
(b) I-1+1-+ +(In
(c) -2 + 4- 8 + 16 + ... + (-2)11
(d) 1.03 + (1.03)2 + (1.03)3 +-... +
(1.03)n
27. Find a rational number representing each of the following repeating decimals:
(a) 0.537537537537537537537537537...
(b) 31.25469696969696969696969...
28. A fixed dose of a given drug increases the concentration of that drug above normal levels in the bloodstream by an amount Co (measured in percent). The effect
of the drug wears off over time such that the concentration at some time t is Coe-kt
where k is the known rate at which the concentration of the drug in the bloodstream
declines.
(a) Find the residual concentration R, the accumulated amount of the drug above normal levels in the bloodstream, at time t after n doses given at intervals of to hours
starting with the first dose at t = 0.
(b) If the drug is alcohol and 1 oz. of alcohol has Co = 0.05%, how often can a "dose"
be taken so that the residual concentration is never more than 0.15%? Assume
k = (1/3) ln(2).
29. (a) Prove by induction that 2n > n for all n > 0.
(b) Prove that 2n > n directly from Theorem 2 in Section 1.7.4, without explicit use of
induction. (That is, Theorem 2 in Section 1.7.4 itself was proved using induction,
but you should not have to do any additional induction.)
(c) Prove by induction that 2" > n
3 for n > 10.
30. Prove by induction:
(a) There is a natural number k such that n! > n
3 for all n > k. (Try to find the least
such number k.)
(b) n! > n
4 for n > 7.
31. Let T = {n E N : sin(n •7r) = 0}. Prove that T = N. (Hint: sin(a + b) = sin(a) •
cos(b) + cos(a) , sin(b).)
32. Prove assertion 1 from Lemma 1.
33. (a) Suppose you take out a mortgage for A dollars at a monthly interest rate I and
a monthly payment P. (To calculate I: if the annual interest rate is 12%, divide
by 12 to get a monthly rate of 1%, then replace the percentage with the decimal
fraction 0.01.) Let An denote the amount you have left to pay off after n months.
So, A 0 = A by definition. At the end of each month, you are first charged interest
CHAPTER 1 Sets, Proof Templates, and Induction
(c) Fo + F 3 +
+ F 3 = F 3 n+ 2 /2 for n > 0
(d)F 1 =Fn "
- (-1)n forn >_ 0
25. The Lucas numbers are defined as LO = 2, L, = 1, and Ln = Ln-1 + Ln-2 for n >
2. Prove the following identities for Lucas numbers.
(a) Lj + L 2 + .. • + Ln = Ln+ 2 - 3 for n > 1
(b) L2
2
2 ...
2-forn>2
(c) L 2 + L 4 + - "- +L 2 n = L 2 n+l - 1 for n > 2
26. Find the value of the following sums:
(a) 2 + 2 -
3n
(b) I-1+1-+ +(In
(c) -2 + 4- 8 + 16 + ... + (-2)11
(d) 1.03 + (1.03)2 + (1.03)3 +-... +
(1.03)n
27. Find a rational number representing each of the following repeating decimals:
(a) 0.537537537537537537537537537...
(b) 31.25469696969696969696969...
28. A fixed dose of a given drug increases the concentration of that drug above normal levels in the bloodstream by an amount Co (measured in percent). The effect
of the drug wears off over time such that the concentration at some time t is Coe-kt
where k is the known rate at which the concentration of the drug in the bloodstream
declines.
(a) Find the residual concentration R, the accumulated amount of the drug above normal levels in the bloodstream, at time t after n doses given at intervals of to hours
starting with the first dose at t = 0.
(b) If the drug is alcohol and 1 oz. of alcohol has Co = 0.05%, how often can a "dose"
be taken so that the residual concentration is never more than 0.15%? Assume
k = (1/3) ln(2).
29. (a) Prove by induction that 2n > n for all n > 0.
(b) Prove that 2n > n directly from Theorem 2 in Section 1.7.4, without explicit use of
induction. (That is, Theorem 2 in Section 1.7.4 itself was proved using induction,
but you should not have to do any additional induction.)
(c) Prove by induction that 2" > n
3 for n > 10.
30. Prove by induction:
(a) There is a natural number k such that n! > n
3 for all n > k. (Try to find the least
such number k.)
(b) n! > n
4 for n > 7.
31. Let T = {n E N : sin(n •7r) = 0}. Prove that T = N. (Hint: sin(a + b) = sin(a) •
cos(b) + cos(a) , sin(b).)
32. Prove assertion 1 from Lemma 1.
33. (a) Suppose you take out a mortgage for A dollars at a monthly interest rate I and
a monthly payment P. (To calculate I: if the annual interest rate is 12%, divide
by 12 to get a monthly rate of 1%, then replace the percentage with the decimal
fraction 0.01.) Let An denote the amount you have left to pay off after n months.
So, A 0 = A by definition. At the end of each month, you are first charged interest