Appendix 1 MAPLE Program for Fibonacci Application

Transcription

1 Appendix 1 MAPLE Program for Fibonacci Application > restart:with(lretools): > eq:¼r(nþ)¼r(nþ1)þr(n); and solved for R(n) subject to the two initial conditions R(0) ¼ 0 and R(1) ¼ N > sol:¼rsolve({eq,r(0)¼0,r(1)¼n},r(n)); > number:¼factor(sol); > N:¼1; n:¼4; > unassign( n ): Springer International Publishing Switzerland 01 E. Grigorieva, Methods of Solving Sequence and Series Problems, DOI /

2 7 Appendix 1 MAPLE Program for Fibonacci Application > Rabbit_pairs:¼radnormal(number); > REplot(eq,R(n),{R(0)¼0,R(1)¼1}, 1..4, labels¼["n","r"], tickmarks¼[3,3], color¼green, thickness¼3); 40,000 R 0, n >

3 Appendix Method of Differences Consider a sequence {a n }, a 1, a, a 3, a 4,... and another sequence, {b n }, b 1, b, b 3, b 4,..., such that the following relationships are valid: a a 1 ¼ b 1 a 3 a ¼ b a 4 a 3 ¼ b 3... a n a n 1 ¼ b n 1 a nþ1 a n ¼ b n : If we add the left and right sides of all equations and denote by S n the n th partial sum of sequence {b n } we obtain the formula, S n ¼ Xn k¼1 b k ¼ a nþ1 a 1 : This formula gives us a straightforward method of finding the n th term of a sequence with integer terms when other methods do not work or perhaps demand too much creativity. Usually one has to subtract the consecutive terms of the obtained difference sequences until a difference sequence contains only the same numbers. Some ideas are demonstrated in the following problems. Problem A1 Given the sequence 1, 5, 15, 35, 70, 1, 10,... find its n th term and the sum of the first n terms. Solution. Consider the series made out of differences of the consecutive terms: 4, 10, 0, 35, 5, 84,..., then a new sequence made out of the difference the consecutive terms of the new sequence, 10, 15, 1, 8,... Finally, the last Springer International Publishing Switzerland 01 E. Grigorieva, Methods of Solving Sequence and Series Problems, DOI /

4 74 Appendix Method of Differences sequence made of the difference of the consecutive terms of 4, 5,, 7,...This is an arithmetic progression with the first term 4 and common difference 1. In general, the solution is similar to Problem 3 or to the homework problem. Let us consider the following sequences: in general it can be written as fa i g 1, 5, 15, 35, 70, 1, 10,... fb i g 4, 10, 0, 35, 5, 84,... fc i g, 10, 15, 1, 8, :::: fd i g 4, 5,, 7, :::: fe 1 g 1, 1, 1, :::: c c 1 ¼ d 1 c 3 c ¼ d c 4 c 3 ¼ d 3... c n c n 1 ¼ d n 1 ; which is equivalent to c n ¼ c 1 þ Xn 1 d n. Because the last difference sequence {d i } forms an arithmetic progression with the first term 4 and common difference of 1, then its n th term can be written as d n ¼ 4 þ ðn 1Þ 1 ¼ n þ 3 and the sum of the first n 1 terms will be 4þ n S n 1 ¼ ð Þ 1 ðn 1Þ ¼ ðnþþðn 1Þ ¼ n þ5n. Therefore, c n ¼ þ n þ5n ¼ n þ5nþ ¼ ðnþþðnþ3þ. This formula as we can see works just fine since c 1 ¼ ð1þþð1þ3þ ¼, c ¼ ðþþðþ3þ ¼ 10, c 3 ¼ ð3þþð3þ3þ ¼ 15, etc. Going up we can now find the second sequence from the top (the first difference) b b 1 ¼ c 1 b 3 b ¼ c b 4 b 3 ¼ c 3... b n b n 1 ¼ c n 1

5 Appendix Method of Differences 75 which is equivalent to b n ¼ b 1 þ Xn 1 c n ¼ 4 þ Xn 1 n þ 5n þ ¼ 4 þ 1 ðn 1Þnðn 1Þ þ 5 nn ð 1Þ þ ðn 1Þ ¼ 4 þ ðn 1Þ ð n þ 7n þ 18Þ ¼ ðn þ 1Þðn þ Þðn þ 3Þ Finally, a n ¼ a 1 þ Xn 1 b n ¼ 1 þ Xn 1 ðn þ 1Þðn þ Þðn þ 3Þ ¼ 1 þ 1 X n 1 n 3 þ Xn 1 n þ 11 X n 1 n þ ðn 1Þ ¼ n þ 1 ðn 1Þ n ð þ n 1 Þn ð n 1 Þ 11 þ ð n 1 Þn 4 1 ¼ nnþ ð 1Þðn þ Þðn þ 3Þ 4 ¼ n4 þ n 3 þ 11n þ n 4 We can see that each term of the given sequence can be evaluated using this formula a 1 ¼ ð Þ ¼ 1, a ¼ ¼ 5,..., a 5 ¼ ¼ 70,... Problem 37 Find the formula for the n th term of the sequence 1, 4, 10, 0, 35, 5, 84, 10,...(See Chapter 1 for geometric interpretation of this problem.) Solution. We can subtract consecutive terms until the difference of the consecutive terms become the same fa i g1, 4, 10, 0, 35, 5, 84, 10,... fb i g 3,, 10, 15, 1, 8, 3,... fc i g 3, 4, 5,, 7, 8,... fd i g 1, 1, 1, 1, 1, ::::: Moving from the last row up, we obtain that

6 7 Appendix Method of Differences a n ¼ a 1 þ Xn 1 b n b n ¼ b 1 þ Xn 1 c n a n ¼ a 1 þ b 1 ðn 1Þþ Xn 1 ð Xn 1 k¼1 The sum inside the parentheses is the ðn 1Þ st partial sum of the arithmetic progression {c i } with the first term 3 and common difference of 1. 3 þ n S n 1 ¼ ðn 1Þ ) b n ¼ 3 þ ðn þ 4Þðn 1Þ a n ¼ 1 þ 1 Xn 1 n 1 þ 3n þ ¼ 1 þ ðn 1Þnðn 1 ¼ 1 þ n 1 1 ðnðn 1Þþ9n þ 1Þ¼ nnþ1 ð Þðn þ Þ c k Þ: ¼ n þ 3n þ Þ þ 3 nn ð 1Þ þ ðn 1Þ Answer. a n ¼ nnþ1 ð ÞðnþÞ :

7 References 1. Williams, K.S., Hardy, K., The Red Book of Mathematics Problems (Undergraduate William Lowell Putnam competition). Dover, Mineola, NY (199). Grigorieva, E.V., Methods of Solving Complex Geometry Problems. Birkhäuser, Basel (013) 3. Grigorieva, E.V., Methods of Solving Nonstandard Problems. Birkhäuser, Basel (015) 4. Grigorieva, E.V., Complex Math Problems and How to Solve Them, vol. 1. TWU Press. Library of Congress, TX u / (001) 5. Dudley, U., Number Theory, nd edn. Dover, Mineola, NY (008). Barton, D., Elementary Number Theory, th edn. McGraw Hill, New York (007) 7. Grigoriev, E. (ed.), Problems of the Moscow State University Entrance Exams, pp MAX-Press, Moscow (00) (In Russian) 8. Grigoriev, E. (ed.), Problems of the Moscow State University Entrance Exams, pp MAX-Press, Moscow (000) (in Russian) 9. Grigoriev, E. (ed.), Olympiads and Problems of the Moscow State University Entrance Exams. MAX-Press, Moscow (008) (in Russian) 10. Wilf, H.S., Generating Functionology. Academic Press, New York (1994) 11. Lidsky, B., Ovsyannikov, L., Tulaikov, A., Shabunin, M., Problems in Elementary Mathematics. MIR Publisher, Moscow (1973) 1. Kaganov, E.D., 400 of the Most Interesting Problems with Solutions. MIR, Moscow (1997) 13. Rivkin, A.A., Problem Book for the Preparation to the Math Entrance Exam. MIR, Moscow (003) 14. Vinogradova, Olehnik, and Sadovnichii, Mathematical Analysis, Vol., Factorial (199) (in Russian) Dunham, W., Euler, The Master of Us All. The Mathematical Association of America, Washington, DC (1999) 17. Eves, J.H., An Introduction to the History of Mathematics with Cultural Connections, pp Harcourt College Publishers, San Diego, CA (1990) 18. Lander, L.J., Parkin, T.R., Consecutive primes in arithmetic progression. Math. Comput. 1, 489 (197) 19. Sierpinski, W., 50 Problems in Elementary Number Theory. Elsevier, New York (1970) 0. Alfutova, N., Ustinov, A., Algebra and Number Theory. MGU, Moscow (009) (in Russian) 1. Currie, L.A., The remarkable metrological history of radiocarbon dating. J. Res. Natl. Inst. Stand. Technol. 109, (004) Springer International Publishing Switzerland 01 E. Grigorieva, Methods of Solving Sequence and Series Problems, DOI /

8 78 References. Anderson, E.C., Libby, W.F., Weinhouse, S., Held, A.F., Kirsohenbaum, A.D., Grosse, A.V., Radiocarbon from cosmic radiation. Science 105, 57 (1947) 3. Grosse, A.F., Libby, W.F., Cosmic radiocarbon and natural radioactivity of living matter. Science 10, 88 (1947) Contest Problems for Further Reading 4. Shklarsky, D.O., Chentzov, N.N., Yaglom, I.M., The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics. Dover, Mineola, NY (1993) 5. Past USAMO tests, The USA Mathematical Olympiads from the late 70s and early 80s can be found at. Art of Problem Solving website Andreescu, T., Feng, Z., Olympiad books. USA and International Mathematical Olympiads (MAA Problem Books Series) 7. Andreescu, T., Feng, Z, Lee, G., Jr. (eds.), Mathematical Olympiads : Problems and Solutions from Around the World. MAA (003) 8. Andreescu, T., Feng, Z. (eds.), Mathematical Olympiads : Problems and Solutions from Around the World. MAA (001)

9 Index A Abel, 153 Abel s method, 177 Abel s Theorem, 149, 153 Absolute error, 150 Absolutely convergent, 14, 143, 145, 150, 151, 4, 1 Accuracy, 150, 151, , 198, 4, 49, 4 Alternating series, 13, 14, , 145, 14, 148, 150, 15, 177, 180, 18, 4, 49, 0, 1 Amortization, 18 Amortization table, 18 Amount of the annuity, 04 Ancient Greeks, 38, 40, 41, 55 Approximation, 14, 137, 138, , 47, 49 Archeology, 194 Archimedes, 5 Arithmetic mean, 31, 101, 104, 108, 0,, Arithmetic progression, 8, 9, 13, 15, 1, 3, 31, 35, 37 41, 47, 49, 50,, 79, 105, 107, , 193, 05, 9 31, 35, 3, 39, 40, 44, 45, 50, 53, 54, 59 Arithmetic series, 4, 7, 50, 5, 1, 175, 37, 39 Artifact, 194 B Babylonians, 4 3, 8 Balance, 03, 05, 08, 09, 5 Balance table, 5 Basic Maclaurin series expansions, 158 Basel problem,, 19, 15, 171, 17 Basic properties of sigma notation, 4 Becquerel, H., 194 Bernoulli,, 19 Binet, Binomial distribution, 159 Borrow money, 13 Boundary, 4 Bounded, 131 C Cantor s Theorem, 11 Carbon 14, 19, 194, 19, 198 Cauchy, 9, 17, , 135, 137, 138, 151, 155, 15, 47, 48, 5 Cauchy-Hadamard formula, 155 Centered hexagonal numbers, 4 Characteristic equation, 30 3, 1 Chemical reaction, 194 Common difference, 8 Common ratio, 17 Comparison tests, 18 Complex numbers, 15 Compounded n times per year, 0 Compound interest, 01 Conditionally convergent, 14, 140, 14, 145, 14, 148, 151, 157, 1 Continuous interest formula, 0 Convergence, 8,, 71, 13 1, 18, 131, 13, 13, 145, 149, 151, 15, , 1, , 181, 33, 47, 48, 0, 1 Springer International Publishing Switzerland 01 E. Grigorieva, Methods of Solving Sequence and Series Problems, DOI /

10 80 Index Convergence of a power series, 153 Credit card/car loan payment, 11 Cubic equation, 1 Curie, M.S., 194 D D Alembert Sufficient Convergence Theorem, 131 De Moivre, A., 93 De Moivre s formula, 9 Derivative, 81, 11, 155, 14 1, 174, 17, 189, 3 Difference of squares, 75, 110, 171, 5, Differential equations, 183 Dirichlet, P.G., 19 Dirichlet series, 19, 130, 138, 140, 145, 14, 150, 170, 48 Dirichlet Theorem, 148 Divergent, 7, 8, 49,,, 70, 79, 80, 98, 99, 13, 1 18, , 14, , 148, 154, 15, 157, 1, 33, 4, 48, 5, 53, 1 Down payments, 0 E Elimination rates, 7 Equivalence of infinitesimals, 13 Error of estimation, 140, 151 Euler, L.,,, 93, 19, 130, 15, 171, 17, 5 Excel, 15, 18 1 Exponential decay, 19 Exponential growth, 193 F Fibonacci rabbits reproduction, 5 Fibonacci sequence, 1,, 3, 4,, 7, 30, 33, 100, 101, 188, 51 Finding an infinite sum, 1 Finite arithmetic progression, 115 Functional series, 13, 151 G Gauss, C.F., 11, 49, 54, , 151 Geiger counter, 194 Generating function, 18 Geometric mean, 101 Geometric progression, 17, 1 3, 7, 31, 33, 101, 105, 113, 139, 181, , 0, 8, 30, 31, 3, 39, 5, 0 Geometric proof, 51, 58, 59 Geometric series, 18, 19, 3, 5,, 9, 78, 81, 95, 99, 10, 110, 17, 143, 154, 155, 17, 174, 185, 18, 188, 199, 04, 0, 10, 13, 3, 3, 40, 41 Graphing calculator,, 19, H Half-lifetime, 195 Harmonic series, 130 Hexagonal number, Homogeneous recursions, 9 I Indiana Jones, 19 Infinite arithmetic progression, 114 Infinite decreasing geometric series, 18 Infinite product, 171 Infinite sums, 14 Integral Convergence Test, 13 Irrational number approximation, 180 K K-polygonal numbers, 4 L Leibniz series, 141, 145, 14, 150, 151, 157, 4 Leibniz Theorem, 141 Leibniz triangle, 8 Limit Comparison Corollary, 18 Loan amortization, 13 Lucas number, 7 M Maclaurin, C., 157 Maclaurin series, 158 MAPLE, 5, 184 Mathematical induction, 9, 34, 49, 50, 9 98, 109, 3, 35, 50, 51, 54, 5 Medication in the blood system, 199 Mengoli, P., 19 Monotonically decreasing, 141

11 Index 81 N Necessary and sufficient, 17 Necessary condition, 15 Net present value of cash flow, 07 Newton, I., 141, 159 Nonnegative numerical series properties, 1 Number theory, 113 Numerical series, 14 O Outstanding balance, 17 Outstanding debt, 13 P Partial sum, Pascal s triangle, 8 Payment scenarios, 08 Pentagonal numbers, 41 Phi and phi, Power series, 13, 15 Present value, 0 Present value of an annuity, 10 Price reduction, 193 Product of two series, 144 Pyramidal number, 45 Q Quadratic equation, 4 Quotient Comparison Theorem, 18 R Radioactive decay, 19, 193, 197 Radioactivity, 194 Radius of convergence, 155 Ratio test, 13 Rationalize the denominator, 78 Reciprocals of triangular numbers, 55 Recurrence, 11, 54 Relatively prime, 10, 117, 9, 53 Repeating decimals, 19 Rewriting fraction as difference, 74 Riemann, B., 148, 151 Root test, 13 Rules of differentiation, 173 S Sequence of consecutive cubes, 3 Sequence of square numbers, 40 Simple interest, 01 Square number, 5 Sum of all natural numbers, 49 Sum of first n triangular numbers, Sum of n cubes, 59 Sum of odd consecutive numbers, 51 Sum of squares of first n natural numbers, Sum of triangular numbers, 5 T Tao, T., 11 Taylor series, 157 Tetrahedral numbers, 44, 45, 5, 88 Theorems for numerical series, 151 Triangular number, 38, 4, 45, 54, 57, 88, 58 Trigonometric series, 9, 15 V Vieta s Theorem, 1