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