 5.5.1: The notationn k=mak is read _____.
 5.5.2: The expanded form ofn k=mak is _____.
 5.5.3: The value of a1 +a2 +a3 ++an when n =2 is _____.
 5.5.4: The notationn k=mak is read _____.
 5.5.5: If n is a positive integer, then n!=_____.
 5.5.6: k=mak +cn k=mbk =_____.
 5.5.7: n k=mak n k=mbk=_____.
 5.5.8: Write the rst four terms of the sequences dened by the formulas. ak...
 5.5.9: Write the rst four terms of the sequences dened by the formulas. bj...
 5.5.10: Write the rst four terms of the sequences dened by the formulas. ci...
 5.5.11: Write the rst four terms of the sequences dened by the formulas. dm...
 5.5.12: Write the rst four terms of the sequences dened by the formulas.Let...
 5.5.13: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.14: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.15: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.16: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.17: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.18: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.19: Find explicit formulas for sequences of the form a1,a2,a3,... with ...
 5.5.20: Let a0 =2, a1 =3, a2 =2, a3 =1, a4 =0, a5 = 1, and a6 =2. Compute e...
 5.5.21: Compute the summations and products k=1(k+1)
 5.5.22: Compute the summations and products 4 k=2k2
 5.5.23: Compute the summations and products 3 m=01 2m
 5.5.24: Compute the summations and products 4 j=0 (1)
 5.5.25: Compute the summations and products 1 i=1i(i +1
 5.5.26: Compute the summations and products 0 j=0(j +1)2j
 5.5.27: Compute the summations and products 2 k=21 1 k
 5.5.28: Compute the summations and products 1 k=1(k2 +3)
 5.5.29: Compute the summations and products 10 n=11 n 1 n+1
 5.5.30: Compute the summations and products i=2i(i +2) (i 1)(i +1)
 5.5.31: Write the summations in expanded form. n i=1(2)i
 5.5.32: Write the summations in expanded form. n j=1j(j +1)
 5.5.33: Write the summations in expanded form.n+1 k=01 k!
 5.5.34: Write the summations in expanded form.k+1 i=1i(i!)
 5.5.35: Evaluate the summations and products for the indicated values of th...
 5.5.36: Evaluate the summations and products for the indicated values of th...
 5.5.37: Evaluate the summations and products for the indicated values of th...
 5.5.38: Evaluate the summations and products for the indicated values of th...
 5.5.39: Rewrite by separating off the nal term.k+1 i=1i(i!)
 5.5.40: Rewrite by separating off the nal term.m+1 k=1k2
 5.5.41: Rewrite by separating off the nal term.n+1 m=1m(m+1)
 5.5.42: Write as a single summation.k i=1i3 +(k+1)3
 5.5.43: Write as a single summation.m k=1k k+1 +m+1 m+2
 5.5.44: Write as a single summation.n m=0(m+1)2m +(n+2)2n+1
 5.5.45: Write using summation or product notation. 12 22 +32 42 +52 62 +72
 5.5.46: Write using summation or product notation. (13 1)(23 1)+(33 1)(43 1...
 5.5.47: Write using summation or product notation. (22 1)(32 1)(42 1)
 5.5.48: Write using summation or product notation. 2 34 3 45 +4 56 5 67 +6 78
 5.5.49: Write using summation or product notation. 1r +r2 r3 +r4 r5
 5.5.50: Write using summation or product notation. (1t)(1t2)(1t3)(1t4)
 5.5.51: Write using summation or product notation. 13 +23 +33 ++n3
 5.5.52: Write using summation or product notation. 1 2!+2 3!+3 4!++n (n+1)!
 5.5.53: Write using summation or product notation. n+(n1)+(n2)++1
 5.5.54: Write using summation or product notation. n+n1 2! +n2 3! +n3 4! ++...
 5.5.55: Transform by making the change of variable i =k+1.5 k=0k(k1)
 5.5.56: Transform by making the change of variable i =k+1.n k=1k k2 +4
 5.5.57: Transform by making the change of variable j =i 1.n+1 i=1(i 1)2 in
 5.5.58: Transform by making the change of variable j =i 1.n i=3i i +n1
 5.5.59: Transform by making the change of variable j =i 1.n1 i=1i (ni)2
 5.5.60: Transform by making the change of variable j =i 1.2n i=nni +1 n+i
 5.5.61: Write as a single summation or product. 3n k=1(2k3)+n k=1(45k)
 5.5.62: Write as a single summation or product. 2n k=1(3k2 +4)+5n k=1(2k2 1)
 5.5.63: Write as a single summation or product. n k=1k k+1 n k=1k+1 k+2
 5.5.64: Compute . Assume the values of the variables are restricted so that...
 5.5.65: Compute . Assume the values of the variables are restricted so that...
 5.5.66: Compute . Assume the values of the variables are restricted so that...
 5.5.67: Compute . Assume the values of the variables are restricted so that...
 5.5.68: Compute . Assume the values of the variables are restricted so that...
 5.5.69: Compute . Assume the values of the variables are restricted so that...
 5.5.70: Compute . Assume the values of the variables are restricted so that...
 5.5.71: Compute . Assume the values of the variables are restricted so that...
 5.5.72: Compute . Assume the values of the variables are restricted so that...
 5.5.73: Compute . Assume the values of the variables are restricted so that...
 5.5.74: Compute . Assume the values of the variables are restricted so that...
 5.5.75: Compute . Assume the values of the variables are restricted so that...
 5.5.76: Compute . Assume the values of the variables are restricted so that...
 5.5.77: Compute . Assume the values of the variables are restricted so that...
 5.5.78: Compute . Assume the values of the variables are restricted so that...
 5.5.79: Compute . Assume the values of the variables are restricted so that...
 5.5.80: Prove that for all nonnegative integers n and r with r +1n, n r +1 ...
 5.5.81: Prove that if p is a prime number and r is an integer with 0 < r < ...
 5.5.82: Mathematical induction is a method for proving that a property dene...
 5.5.83: Let P(n) be a property dened for integers n and consider constructi...
 5.5.84: Use mathematical induction (and the proof of Proposition 5.2.1 as a...
 5.5.85: Use mathematical induction to show that any postage of at least 12c...
 5.5.86: For each positive integer n, letP(n) be the formula 12 +22 ++n2 = n...
 5.5.87: For each integer n with n 2, let P(n) be the formula n1 i=1 i(i +1)...
 5.5.88: Fill in the missing pieces in the following proof that 1+3+5++(2n1)...
 5.5.89: Prove using mathematical induction. Do not derive them from Theorem...
 5.5.90: Prove using mathematical induction. Do not derive them from Theorem...
 5.5.91: Prove using mathematical induction. Do not derive them from Theorem...
 5.5.92: Prove using mathematical induction. Do not derive them from Theorem...
 5.5.93: Prove by mathematical induction. 12 +22 ++n2 =n(n+1)(2n+1) 6, for a...
 5.5.94: Prove by mathematical induction. 13 +23 ++n3 =n(n+1) 22, for all in...
 5.5.95: Prove by mathematical induction. 12 +1 23 ++1 n(n+1) =n n+1, for al...
 5.5.96: Prove by mathematical induction. n1 i=1i(i +1) =n(n1)(n+1) 3, for a...
 5.5.97: Prove by mathematical induction. n+1 i=1i2i =n2n+2 +2, for all inte...
 5.5.98: Prove by mathematical induction. n i=1i(i!) = (n+1)!1, for all inte...
 5.5.99: Prove by mathematical induction. 1 1 221 1 321 1 n2= n+1 2n, for al...
 5.5.100: Prove by mathematical induction. n i=0 1 2i +11 2i +2= 1 (2n+2)!, f...
 5.5.101: If x is a real number not divisible by , then for all integers n 1,...
 5.5.102: (For students who have studied calculus) Use mathematical induction...
 5.5.103: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.104: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.105: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.106: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.107: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.108: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.109: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.110: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.111: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.112: Usetheformulaforthesumoftherstn integersand/ortheformula for the su...
 5.5.113: Find a formula in n,a,m, andd for the sum (a+md)+ (a+(m+1)d)+(a+(m+...
 5.5.114: Find a formula in a,r,m, andn for the sum arm +arm+1 +arm+2 ++arm+n...
 5.5.115: You have two parents, four grandparents, eight greatgrandparents, a...
 5.5.116: Find the mistakes in the proof fragments . Theorem: For any integer...
 5.5.117: Find the mistakes in the proof fragments . Theorem: For any integer...
 5.5.118: Find the mistakes in the proof fragments . Theorem: For any integer...
 5.5.119: Use Theorem 5.2.2 to prove that if m and n are any positive integer...
 5.5.120: Use Theorem 5.2.2 and the result of exercise 10 to prove that if p ...
 5.5.121: Mathematical induction differs from the kind of induction used in t...
 5.5.122: Mathematical induction can be used to _____ conjectures that have b...
 5.5.123: Based on the discussion of the product 1 1 21 1 3 1 1 41 1 n at the...
 5.5.124: Experiment with computing values of the product 1+ 1 11+ 1 21+ 1 31...
 5.5.125: Observe that1 13 =1 31 13 +1 35 =2 51 13 +1 35 +1 57 =3 71 13 +1 35...
 5.5.126: Observe that1=1, 14= (1+2), 14+9=1+2+3, 14+916= (1+2+3+4), 14+916+2...
 5.5.127: Evaluate the sumn k=1k (k+1)!for n =1,2,3,4, and 5. Make a conjectu...
 5.5.128: For each positive integer n, letP(n) be the property 5n 1 is divisi...
 5.5.129: For each positive integer n, letP(n) be the property2n <( n+1)!. a....
 5.5.130: Prove each statement by mathematical induction. 5n 1 is divisible b...
 5.5.131: Prove each statement by mathematical induction. 7n 1 is divisible b...
 5.5.132: Prove each statement by mathematical induction. n3 7n+3 is divisibl...
 5.5.133: Prove each statement by mathematical induction. n3 7n+3 is divisibl...
 5.5.134: Prove each statement by mathematical induction. For any integer n 0...
 5.5.135: Prove each statement by mathematical induction. For any integer n 0...
 5.5.136: Prove each statement by mathematical induction. n3 n is divisible b...
 5.5.137: Prove each statement by mathematical induction. n(n2 +5) is divisib...
 5.5.138: Prove each statement by mathematical induction. 2n <( n+1)!, for al...
 5.5.139: Prove each statement by mathematical induction. 1+3n 4n, for every ...
 5.5.140: Prove each statement by mathematical induction. 5n +9 < 6n, for all...
 5.5.141: Prove each statement by mathematical induction. n2 < 2n, for all in...
 5.5.142: Prove each statement by mathematical induction. 2n <( n+2)!, for al...
 5.5.143: Prove each statement by mathematical induction. n < 1 1 +1 2 ++1 n,...
 5.5.144: Prove each statement by mathematical induction. 1+nx (1+x)n, for al...
 5.5.145: Prove each statement by mathematical induction. a. n3 > 2n+1, for a...
 5.5.146: A sequence a1,a2,a3,...is dened by letting a1 =3 and ak =7ak1 for a...
 5.5.147: A sequence b0,b1,b2,...is dened by letting b0 =5 and bk =4+bk1 fora...
 5.5.148: A sequence c0,c1,c2,...is dened by letting c0 =3 and ck = (ck1)2 fo...
 5.5.149: A sequence d1,d2,d3,...is dened by letting d1 =2 and dk = dk1 k for...
 5.5.150: Prove that for all integers n 1, 1 3 = 1+3 5+7 = 1+3+5 7+9+11 = = 1...
 5.5.151: As each of a group of businesspeople arrives at a meeting, each sha...
 5.5.152: In order for a proof by mathematical induction to be valid, the bas...
 5.5.153: In order for a proof by mathematical induction to be valid, the bas...
 5.5.154: Some55checkerboardswithonesquareremovedcanbe completely covered by ...
 5.5.155: Consider a 46 checkerboard. Draw a covering of the board by Lshape...
 5.5.156: a. Use mathematical induction to prove that any checkerboard with d...
 5.5.157: Let m and n be any integers that are greater than or equal to 1. a....
 5.5.158: In a roundrobin tournament each team plays every other team exactl...
 5.5.159: On the outside rim of a circular disk the integers from 1 through 3...
 5.5.160: Suppose that nas and nbs are distributed around the outside of a ci...
 5.5.161: For a polygon to be convex means that given any two points on or in...
 5.5.162: a. Prove that in an 88 checkerboard with alternating black and whit...
 5.5.163: In a proof by strong mathematical induction the basis step mayrequi...
 5.5.164: Suppose that in the basis step for a proof by strong mathematical i...
 5.5.165: Accordingtothewellorderingprinciplefortheintegers,ifa set S of int...
 5.5.166: Suppose a1,a2,a3,...is a sequence dened as follows: a1 =1, a2 =3, a...
 5.5.167: Suppose b1,b2,b3,...is a sequence dened as follows: b1 =4, b2 =12 b...
 5.5.168: Supposethatc0,c1,c2,...isasequencedenedasfollows: c0 =2, c1 =2, c2 ...
 5.5.169: Supposethatd1,d2,d3,...isasequencedenedasfollows: d1 = 9 10 , d2 = ...
 5.5.170: Supposethate0,e1,e2,...isasequencedenedasfollows: e0 =12, e1 =29 ek...
 5.5.171: Suppose that f0, f1, f2,... is a sequence dened as follows: f0 =5, ...
 5.5.172: Suppose that g1,g2,g3,... is a sequence dened as follows: g1 =3, g2...
 5.5.173: Suppose that h0,h1,h2,... is a sequence dened as follows: h0 =1, h1...
 5.5.174: Deneasequencea1,a2,a3,...asfollows:a1 =1,a2 =3, andak =ak1 +ak2 for...
 5.5.175: The problem that was used to introduce ordinary mathematical induct...
 5.5.176: You begin solving a jigsaw puzzle by nding two pieces that match an...
 5.5.177: The sides of a circular track contain a sequence of cans of gasolin...
 5.5.178: Use strong mathematical induction to prove the existence partoftheu...
 5.5.179: Any product of two or more integers is a result of successive multi...
 5.5.180: Dene the sum of one integer to be that integer, and use strong math...
 5.5.181: Use strong mathematical induction to prove that for any integer n 2...
 5.5.182: Compute 41,42,43,44,45,46,47, and 4 8. Make a conjecture about the ...
 5.5.183: Compute90,91,92,93,94,and9 5.Makeaconjectureabout theunitsdigitof9n...
 5.5.184: Find the mistake in the following proof that purports to show that ...
 5.5.185: Use the wellordering principle for the integers to prove Theorem 4...
 5.5.186: Use the wellordering principle for the integers to prove the exist...
 5.5.187: a. The Archimedean property for the rational numbers states that fo...
 5.5.188: Use the results of exercise 22 and the wellordering principle for ...
 5.5.189: Usethewellorderingprincipletoprovethatgivenanyinteger n 1, there e...
 5.5.190: Imagine a situation in which eight people, numbered consecutively 1...
 5.5.191: Suppose P(n) is a property such that 1. P(0), P(1), P(2) are all tr...
 5.5.192: Prove that if a statement can be proved by strong mathematical indu...
 5.5.193: Give examples to illustrate the proof of Theorem 5.4.1.
 5.5.194: Write the following numbers in decimal notation: a.11102 b.101112 c...
 5.5.195: It is a fact that every integer n 1 can be written in the form cr3r...
 5.5.196: Use mathematical induction to prove the existence part of the quoti...
 5.5.197: Prove that if a statement can be proved by ordinary mathematical in...
 5.5.198: Use the principle of ordinary mathematical induction to prove the w...
 5.5.199: A recursive denition for a sequence consists of a _____ and _____.
 5.5.200: A recurrence relation is an equation that denes each later term of ...
 5.5.201: Initialconditionsforarecursivedenitionofasequenceconsist of one or ...
 5.5.202: To solve a problem recursively means to divide the problem into sma...
 5.5.203: A crucial step for solving a problem recursively is to dene a _____...
 5.5.204: Find the rst four terms of each of the recursively dened sequences....
 5.5.205: Find the rst four terms of each of the recursively dened sequences....
 5.5.206: Find the rst four terms of each of the recursively dened sequences....
 5.5.207: Find the rst four terms of each of the recursively dened sequences....
 5.5.208: Find the rst four terms of each of the recursively dened sequences....
 5.5.209: Find the rst four terms of each of the recursively dened sequences....
 5.5.210: Find the rst four terms of each of the recursively dened sequences....
 5.5.211: Find the rst four terms of each of the recursively dened sequences....
 5.5.212: Find the rst four terms of each of the recursively dened sequences....
 5.5.213: Letb0,b1,b2,...bedenedbytheformulabn =4n,forall integers n 0. Show ...
 5.5.214: Let c0,c1,c2,... be dened by the formula cn =2n 1 for all integers ...
 5.5.215: Let s0,s1,s2,... be dened by the formula sn =(1)n n!for all integer...
 5.5.216: Let t0,t1,t2,...be dened by the formula tn =2+n for all integers n ...
 5.5.217: Let d0,d1,d2,...be dened by the formula dn =3n 2n for all integers ...
 5.5.218: For the sequence of Catalan numbers dened in Example 5.5.4, prove t...
 5.5.219: Use the recurrence relation and values for the Tower of Hanoi seque...
 5.5.220: TowerofHanoiwithAdjacencyRequirement:Supposethat inadditiontothereq...
 5.5.221: Tower of Hanoi with Adjacency Requirement: Suppose the same situati...
 5.5.222: FourPole Tower of Hanoi: Suppose that the Tower of Hanoi problem h...
 5.5.223: Tower of Hanoi Poles in a Circle: Suppose that instead of being lin...
 5.5.224: Double Tower of Hanoi: In this variation of the Tower of Hanoi ther...
 5.5.225: Fibonacci Variation: A single pair of rabbits (male and female) is ...
 5.5.226: Fibonacci Variation: A single pair of rabbits (male and female) is ...
 5.5.227: F0, F1, F2,...is the Fibonacci sequence. Use the recurrence relatio...
 5.5.228: F0, F1, F2,...is the Fibonacci sequence. The Fibonacci sequence sat...
 5.5.229: F0, F1, F2,...is the Fibonacci sequence.Prove that Fk =3Fk3 +2Fk4 f...
 5.5.230: F0, F1, F2,...is the Fibonacci sequence.Prove that F2 k F2 k1 = FkF...
 5.5.231: F0, F1, F2,...is the Fibonacci sequence.Prove that F2 k+1 F2 k F2 k...
 5.5.232: F0, F1, F2,...is the Fibonacci sequence.Prove that F2 k+1 F2 k = Fk...
 5.5.233: F0, F1, F2,...is the Fibonacci sequence.Use mathematical induction ...
 5.5.234: F0, F1, F2,...is the Fibonacci sequence.Use strong mathematical ind...
 5.5.235: F0, F1, F2,...is the Fibonacci sequence.It turns out that the Fibon...
 5.5.236: F0, F1, F2,...is the Fibonacci sequence. (For students who have stu...
 5.5.237: F0, F1, F2,...is the Fibonacci sequence. (For students who have stu...
 5.5.238: (For students who have studied calculus) Dene x0,x1,x2,...as follow...
 5.5.239: Compound Interest: Suppose a certain amount of money is deposited i...
 5.5.240: Compound Interest: Suppose a certain amount of money is deposited i...
 5.5.241: Witheachstepyoutakewhenclimbingastaircase,youcan move up either one...
 5.5.242: A set of blocks contains blocks of heights 1, 2, and 4 centimeters....
 5.5.243: Use the recursive denition of summation, together with mathematical...
 5.5.244: Usetherecursivedenitionofproduct,togetherwithmathematical induction...
 5.5.245: Usetherecursivedenitionofproduct,togetherwithmathematical induction...
 5.5.246: The triangle inequality for absolute value states that for all real...
 5.5.247: To use iteration to nd an explicit formula for a recursively dened ...
 5.5.248: At every step of the iteration process, it is important to eliminat...
 5.5.249: If a single number, say a, is added to itself k times in one of the...
 5.5.250: If a single number, say a, is multiplied by itself k times in one o...
 5.5.251: A general arithmetic sequence a0, a1, a2,... with initial value a0 ...
 5.5.252: A general geometric sequence a0, a1, a2,... with initial value a0 a...
 5.5.253: Whenanexplicitformulaforarecursivelydenedsequence has been obtained...
 5.5.254: The formula1+2+3++n =n(n+1) 2 is true for all integers n 1. Use thi...
 5.5.255: The formula1+r +r2 ++rn =rn+1 1 r 1is true for all real numbers r e...
 5.5.256: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.257: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.258: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.259: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.260: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.261: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.262: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.263: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.264: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.265: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.266: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.267: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.268: In a sequence is dened recursively. Use iteration to guess an expli...
 5.5.269: Solve the recurrence relation obtained as the answer to exercise 17...
 5.5.270: Solve the recurrence relation obtained as the answer to exercise 21...
 5.5.271: Supposed isaxedconstantanda0,a1,a2,...isasequence that satises the ...
 5.5.272: A worker is promised a bonus if he can increase his productivity by...
 5.5.273: A runner targets herself to improve her time on a certain course by...
 5.5.274: Suppose r is a xed constant and a0,a1,a2 ...is a sequence that sati...
 5.5.275: As shown in Example 5.5.8, if a bank pays interest at a rate of i c...
 5.5.276: Supposethepopulationofacountryincreasesatasteadyrate of 3% per year...
 5.5.277: A chain letter works as follows: One person sends a copy of the let...
 5.5.278: A certain computer algorithm executes twice as many operations when...
 5.5.279: A person saving for retirement makes an initial deposit of $1,000 t...
 5.5.280: A person borrows $3,000 on a bank credit card at a nominal rate of ...
 5.5.281: Use mathematical induction to verify the correctness of the formula...
 5.5.282: Use mathematical induction to verify the correctness of the formula...
 5.5.283: Use mathematical induction to verify the correctness of the formula...
 5.5.284: Use mathematical induction to verify the correctness of the formula...
 5.5.285: Use mathematical induction to verify the correctness of the formula...
 5.5.286: Use mathematical induction to verify the correctness of the formula...
 5.5.287: Use mathematical induction to verify the correctness of the formula...
 5.5.288: Use mathematical induction to verify the correctness of the formula...
 5.5.289: Use mathematical induction to verify the correctness of the formula...
 5.5.290: Use mathematical induction to verify the correctness of the formula...
 5.5.291: Use mathematical induction to verify the correctness of the formula...
 5.5.292: Use mathematical induction to verify the correctness of the formula...
 5.5.293: Use mathematical induction to verify the correctness of the formula...
 5.5.294: Use mathematical induction to verify the correctness of the formula...
 5.5.295: Use mathematical induction to verify the correctness of the formula...
 5.5.296: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.297: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.298: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.299: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.300: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.301: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.302: Inasequenceisdenedrecursively.(a)Useiterationtoguessanexplicitformu...
 5.5.303: Determine whether the given recursively dened sequence satises the ...
 5.5.304: Determine whether the given recursively dened sequence satises the ...
 5.5.305: A single line divides a plane into two regions. Two lines (by cross...
 5.5.306: Compute11 10 n for small values of n (up to about 5 or 6). Conjectu...
 5.5.307: Ineconomicsthebehaviorofaneconomyfromoneperiodto another is often m...
Solutions for Chapter 5: Sequences, Mathematical Induction, and Recursion
Full solutions for Discrete Mathematics: Introduction to Mathematical Reasoning  1st Edition
ISBN: 9780495826170
Solutions for Chapter 5: Sequences, Mathematical Induction, and Recursion
Get Full SolutionsSince 307 problems in chapter 5: Sequences, Mathematical Induction, and Recursion have been answered, more than 17079 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics: Introduction to Mathematical Reasoning was written by and is associated to the ISBN: 9780495826170. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics: Introduction to Mathematical Reasoning, edition: 1. Chapter 5: Sequences, Mathematical Induction, and Recursion includes 307 full stepbystep solutions.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.