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n! > nº, for all inte For all integers n 1 3 II

n! > nº, for all inte For all integers n 1 3 II

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve the question no 2 part a in picture

As according to policy u solve three parts for a question solve all three parts in detail step by step

Discrete mathematics 

Q#1 (a): Write negations, contrapositive, converse and inverse for each statement. (i) If p is a square, then p is a rectangle. (ii) If the decimal expansion of r is terminating, then r is rational. (b): Construct truth table to verify the following equivalences. (i) p → (q v r) = (p ^~ q) → r. (ii) p ^ (q V r) = (p ^ q) v (p ^ r). (c): Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. (i) If this number is greater than 2, then its square is greater than 4. This number is not greater than 2. .. The square of this number is not greater than 4. (ii) If this computer program is correct, then it produces the correct output when run with the test data my teacher gave me. The computer program produces the correct output when run with the test data my teacher gave me. .. This computer program is correct. Q#2 (a): Use mathematical induction to show that: (i) n! > n², for all integers n 2 4. (ii) For all integers n > 1, 1+3 35+7 (iii) n(n² + 5) is divisible by 6, V n 2 0. 1 1+3+5 1+3+ ...+ (2n – 1) 7+ 9 + 11 (2n + 1) + ..+ (4n – 1) (b): Fibonacci Variation: A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: (1) Rabbit pairs are not fertile during the first month of their life but thereafter give birth to four new male/female pair at the end of every month. (2) No rabbits die. (i) Let r, = the number of pairs of rabbits alive at the end of month n, for each integer n 2 1 and let ro = 1. Find a recurrence relation for r1,r2,r3, ... (ii) How many rabbits will be there at the end of the year?
Transcribed Image Text:Q#1 (a): Write negations, contrapositive, converse and inverse for each statement. (i) If p is a square, then p is a rectangle. (ii) If the decimal expansion of r is terminating, then r is rational. (b): Construct truth table to verify the following equivalences. (i) p → (q v r) = (p ^~ q) → r. (ii) p ^ (q V r) = (p ^ q) v (p ^ r). (c): Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. (i) If this number is greater than 2, then its square is greater than 4. This number is not greater than 2. .. The square of this number is not greater than 4. (ii) If this computer program is correct, then it produces the correct output when run with the test data my teacher gave me. The computer program produces the correct output when run with the test data my teacher gave me. .. This computer program is correct. Q#2 (a): Use mathematical induction to show that: (i) n! > n², for all integers n 2 4. (ii) For all integers n > 1, 1+3 35+7 (iii) n(n² + 5) is divisible by 6, V n 2 0. 1 1+3+5 1+3+ ...+ (2n – 1) 7+ 9 + 11 (2n + 1) + ..+ (4n – 1) (b): Fibonacci Variation: A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: (1) Rabbit pairs are not fertile during the first month of their life but thereafter give birth to four new male/female pair at the end of every month. (2) No rabbits die. (i) Let r, = the number of pairs of rabbits alive at the end of month n, for each integer n 2 1 and let ro = 1. Find a recurrence relation for r1,r2,r3, ... (ii) How many rabbits will be there at the end of the year?
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