diff --git a/chapter_5/exercises.md b/chapter_5/exercises.md index 43232a2..0281519 100644 --- a/chapter_5/exercises.md +++ b/chapter_5/exercises.md @@ -8617,3 +8617,264 @@ Omitted. of $n$ odd integers that is even, then $n$ is even. Omitted. + +--- + +Page 373 + +**Exercise Set 5.7** + +1. The formula + +$$ 1 + 2 + 3 + \dots + n = \frac{n(n + 1)}{2} $$ + +is true for every integer $n \geq 1$. Use this fact to solve each of the +following problems: + +a. If $k$ is an integer and $k \geq 2$, find a formula for the expression +$1 + 2 + 3 + \dots + (k - 1)$. + +b. If $n$ is an integer and $n \geq 1$, find a formula for the expression +$5 + 2 + 4 + 6 + 8 + \dots + 2n$. + +c. If $n$ is an integer and $n \geq 1$, find a formula for the expression +$3 + 3 \cdot 2 + 3 \cdot 3 + \dots + 3 \cdot n + n$. + +2. The formula + +$$ 1 + r + r^2 + \dots + r^n = \frac{r^{n + 1} - 1}{r - 1} $$ + +is true for every real number $r$ except $r = 1$ and for every integer +$n \geq 0$. Use this fact to solve each of the following problems: + +a. If $i$ is an integer and $i \geq 1$, find a formula for the expression +$1 + 2 + 2^2 + \dots + 2^{i - 1}$. + +b. If $n$ is an integer and $n \geq 1$, find a formula for the expression +$3^{n - 1} + 3^{n - 2} + \dots + 3^2 + 3 + 1$. + +c. If $n$ is an integer and $n \geq 2$, find a formula for the expression +$2^n + 2^{n - 2} \cdot 3 + 2^{n - 3} \cdot 3 + \dots + 2^2 \cdot 3 + 2 \cdot 3 + 3$. + +d. If $n$ is an integer and $n \geq 1$, finda formula for the expression + +$$ 2^n - 2^{n - 1} + 2^{n - 2} - 2^{n - 3} + \dots + (-1)^{n - 1} \cdot 2 + (-1)^n $$ + +In each of 3-15 a sequence is defined recursively. Use iteration to guess an +explicit formula for the sequence. Use formulas from Section 5.2 to simplify +your answers whenever possible. + +3. $a_k = ka_{k - 1}$, for each integer $k \geq 1$ $a_0 = 1$. + +4. $b_k = \dfrac{b_{k - 1}}{1 + b_{k - 1}}$, for each integer $k \geq 1$ + $b_0 = 1$. + +5. $c_k = 3c_{k - 1} + 1$, for each integer $k \geq 2$ $c_1 = 1$. + +6. $d_k =2d_{k j 1} + 3$, for each integer $k \geq 2$, $d_1 = 2$. + +7. $e_k = 4e_{k - 1} + 5$, for each integer $k \geq 1$ $e_0 = 2$. + +8. $f_k = f_{k - 1} + 2^k$, for each integer $k \geq 2$ $f_1 = 1$. + +9. $g_k = \dfrac{g_{k - 1}}{g_{k - 1} + 2}$, for each integer $k \geq 2$ + $g_1 = 1$. + +10. $h_k = 2^k - h_{k - 1}$, for each integer $k \geq 1$ $h_0 = 1$. + +11. $p_k, = p_{k - 1} + 2 \cdot 3^k$, for each integer $k \geq 2$ $p_1 = 2$. + +12. $s_k = s_{k - 1} + 2k$, for each integer $k \geq 1$ $s_0 = 3$. + +13. $t_k = t_{k - 1} + 3k + 1$, for each integer $k \geq 1$ $t_0 = 0$. + +14. $x_k = 3x_{k - 1} + k$, for each integer $k \geq 2$ $x_1 = 1$. + +15. $y_k = y_{k - 1} + k^2$, for each integer $k \geq 2$ $y_1 = 1$. + +16. Solve the recurrence relation obtained as the answer to exercise 17\(c\) of + Section 5.6. + +17. Solve the recurrence relation obtained as the answer to exercise 21\(c\) of + Section 5.6. + +18. Suppose $d$ is a fixed constant and $a_0, a_1, a_2, \dots$ is a sequence + that satisfies the recurrence relation $a_k = a_{k - 1} + d$, for each + integer $k \geq 1$. Use mathematical induction to prove that + $a_n = a_0 + nd$, for every integer $n \geq 0$. + +19. A worker is promised a bonus if he can increase his productivity by 2 units + a day for a period of 30 days. If on day 0 he produces 170 units, how many + units must he produce on day 30 to qualify for the bonus? + +20. A runner targets herself to improve her time on a certain course by 3 + seconds a day. If on day 0 she runs the course in 3 minutes, how fast must + she run it on day 14 to stay on target? + +21. Suppose $r$ is a fixed constant and $a_0, a_1, a_2, \dots$ is a sequence + that satisfies the recurrence 4elation $a_k = ra_{k - 1}$, for each integer + $k \geq 1$ and $a_0 = a$. Use mathematical induction to prove that + $a_n = ar^n$, for every integer $n \geq 0$. + +22. As shown in Example 5.6.8, if a bank pays interest at a rate of $i$ + compounded $m$ times a year, then the amount of money $P_k$ at the end of + $k$ time periods (where one time period = $\dfrac{1}{m}$th of a + year) satisfies the recurrence relation + $P_k = \left[1 + \left(\dfrac{1}{m}\right)\right]P_{k - 1}$ with initial + condition $P_0 = \text{ the initial amount deposited}$. Find an explicit + formula for $P_n$. + +23. Suppose the population of a country increases at a steady rate of 3% per + year. If the population is 50 million at a certain time, what will it be 25 + years later? + +24. A chain letter works as follows: One person sends a copy of the letter to + five friends, each of whom sends a copy to five friends, each of whom sends + a copy to five friends, each of whom sends a copy to five friends, and so + forth. How many people will have received copies of the letter after the + twentieth reception of this process, assuming no person receives more than + one copy? + +25. A certain computer algorithm executes twice as many operations when it is + run with an input size $k$ as when it is run with an input size $k - 1$ + (where $k$ is an integer that is greater than $1$). When the algorithm is + run with an input size $1$, it executes seven operations. How many + operations does it execute when it is run with an input size of $25$? + +26. A person saving for retirement makes an initial deposit of $1,000 to a bank + account earning interest at a rate of 3% per year compounded monthly, and + each month she adds an addition $200 to the account. + +a. For each nonnegative integer $n$, let $A_n$ be the amount in the account at +the end of $n$ months. Find the recurrence relation relating $A_k$ to +$A_{k - 1}$. + +b. Use iteration to find an explicit formula for $A_n$. + +c. Use mathematical induction to prove the correctness of the formula you +obtained in part (b). + +d. How much will the account be worth at the end of 20 years? At the end of 40 +years? + +e. In how many years will the account be worth $10,000? + +27. A person borrows $3,000 on a bank credit card at a nominal rate of 18% per + year, which is actually charged at a rate of 1.5% per month. + +a. What is the annual percentage yield (APY) for the card? (See Example 5.6.8 +for a definition of APY.) + +b. Assume that the person does not place any additional charges on the card and +pays the bank $150 each month to pay off the loan. Let $B_n$ be the balance owed +on the card after $n$ months. Find an explicit formula for $B_n$. + +c. How long will be required to pay off the debt? + +d. What is the total amount of money the person will have paid for the loan? + +In 28-42 use mathematical induction to verify the correctness of the formula you +obtained in the referenced exercise. + +28. Exercise 3 + +29. Exercise 4 + +30. Exercise 5 + +31. Exercise 6 + +32. Exercise 7 + +33. Exercise 8 + +34. Exercise 9 + +35. Exercise 10 + +36. Exercise 11 + +37. Exercise 12 + +38. Exercise 13 + +39. Exercise 14 + +40. Exercise 15 + +41. Exercise 16 + +42. Exercise 17 + +In each of 43-49 a sequence is defined recursively. (a) Use iteration to guess +an explicit formula for the sequence. (b) Use strong mathematical induction to +verify that the formula of part (a) is correct. + +43. $a_k = \dfrac{a_{k - 1}}{2a_{k - 1} - 1}$, for each integer $k \geq 1$ + $a_0 = 2$. + +44. $b_k = \dfrac{2}{b_{k - 1}}$, for each integer $k \geq 2$ $b_1 = 1$. + +45. $v_k = v_{\lfloor \dfrac{k}{2} \rfloor} + v_{\lfloor \dfrac{(k + 1)}{2}\rfloor} + 2$, + for each integer $k \geq 2$ $v_1 = 1$. + +46. $s_k = 2s_{k - 2}$, for each integer $k \geq 2$ $s_0 = 1$, $s_1 = 2$. + +47. $t_k = k - t_{k - 1}$, for each integer $k \geq 1$ $t_0 = 0$. + +48. $w_k = w_{k - 2} + k$, for each integer $k \geq 3$ $w_1 = 1$, $w_2 = 2$. + +49. $u_k = u_{k - 2} \cdot u_{k - 1}$, for each integer $k \geq 2$ + $u_0 = u_1 = 2$ + +In 50 and 51 determine whether the given recursively defined sequence satisfies +the explicit formula $a_n = (n - 1)^2$, for every integer $n \geq 1$. + +50. $a_k = 2a_{k - 1} + k - 1$, for each integer $k \geq 2$ $a_1 = 0$. + +51. $a_k = 4a_{k - 1} - k + 3$, for each integer $k \geq 2$ $a_1 = 0$. + +52. A single line divides a plane into two regions. Two lines (by crossing) can + divide a plane into four regions; three lines can divide it into seven + regions (see the figure). Let $P_n$ be the maximum number of regions into + which $n$ lines divide a plane, where $n$ is a positive integer. + +[See Page 375 for image] + +a. Derive a recurrence relation for $P_k$ in terms of $P_{k - 1}$, for each +integer $k \geq 2$. + +b. Use iteration to guess an explicit formula for $P_n$. + +53. Compute $\left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array}\right]^n$ for + small values of $n$ (up to about 5 or 6). Conjecture explicit formulas for + the entries in this matrix, and prove your conjecture using mathematical + induction. + +54. In economics the behavior of an economy from one period to another is often + modeled by recurrence relations. Let $Y_k$ be the income in period $k$ and + $C_k$ be the consumption in period $k$. In one economic model, income in any + period is assumed to be the sum of consumption in that period plus + investment and government expenditures (which are assumed to be constant + from period to period), and consumption in each period is assumed to be a + linear function of the income of the preceding period. That is, + +$$ Y_k = C_k + E $$ + +where $E$ is the sum of investment plus government expenditures. + +$$ C_k = c + mY_{k - 1} $$ + +where $c$ and $m$ are constants. + +Substituting the second equation into the first gives +$Y_k = E + c + mY_{k - 1}$. + +a. Use iteration on the above recurrence relation to obtain + +$$ Y_n = (E + c)\left(\frac{m^n - 1}{m - 1}\right) + m^nY_0 $$ + +for every integer $n \geq 1$. + +b. (For students who have studied calculus) Show that if $0 < m < 1$, then +$\lim\limits_{m \to \infty}Y_n = \dfrac{E + c}{1 - m}$. diff --git a/chapter_5/notes.md b/chapter_5/notes.md index 273437e..9d11a16 100644 --- a/chapter_5/notes.md +++ b/chapter_5/notes.md @@ -1000,3 +1000,33 @@ The **product from $i = 1$ to $n$ of the $a_i$**, denoted $\prod_{i = 1}^{n}{a_i}$, is defined by $$ \prod_{i = 1}^{1}{a_i} = a_1 \quad \text{ and } \quad \prod_{i = 1}^{n}{a_i} = \left(\prod_{i = 1}^{n - 1}{a_i}\right) \cdot a_n, \quad \text{ if } \quad n > 1 $$ + +--- + +Page 365 + +**Definition** + +A sequence $a_0, a_1, a_2, \dots$ is called an **arithmetic sequence** if, and +only if, there is a constant $d$ such that + +$$ a_k = a_{k - 1} + d \quad \text{ for each integer } k \geq 1 $$ + +It follows that + +$$ a_n = a_0 + dn \quad \text{ for every integer } n \geq 0 $$ + +--- + +Page 366 + +**Definition** + +A sequence $a_0, a_1, a_2, \dots$ is called a **geometric sequence** if, and +only if, there is a constant $r$ such that + +$$ a_k = ra_{k - 1} \quad \text{ for each integer } k \geq 1 $$ + +It follows that + +$$ a_n = a_0r^n \quad \text{ for each integer } n \geq 0 $$ diff --git a/chapter_5/test_yourself.md b/chapter_5/test_yourself.md index 256b2a3..5149637 100644 --- a/chapter_5/test_yourself.md +++ b/chapter_5/test_yourself.md @@ -175,3 +175,32 @@ that the smaller subproblems have already been solved; solve the initial problem specified. sequence + +--- + +Page 372 + +**Test Yourself** + +1. To use iteration to find an explicit formula for a recursively defined + sequence, start with the _____ and use successive substitution into the _____ + to look for a numerical pattern. + +2. At every step of the iteration process, it is important to eliminate _____. + +3. If a single number, say $a$, is added to itself $k$ times in one of the steps + of the iteration, replace the sum by the expression _____. + +4. If a single number, say $a$, is multiplied by itself $k$ times in one of the + steps of the iteration, replace the product by the expression _____. + +5. A general arithmetic sequence $a_0, a_1, a_2, \dots$ with initial value $a_0$ + and fixed constant summand $d$ satisfies the recurrence relation _____ and + has the explicit formula _____. + +6. A general geometric sequence $a_0, a_1, a_2, \dots$ with initial value $a_0$ + and fixed constant multiplier $r$ satisfies the recurrence relation _____ and + has the explicit formula _____. + +7. When an explicit formula for a recursively defined sequence has been obtained + by iteration, its correctness can be checked by _____.