CALCULUS W/SAPLING ACCESS >IC<
4th Edition
ISBN: 9781319323394
Author: Rogawski
Publisher: MAC HIGHER
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Question
Chapter C, Problem 2E
To determine
To prove:
The formula forall natural number using the principle of induction.
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Chapter C Solutions
CALCULUS W/SAPLING ACCESS >IC<
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- Prove by induction that 1+2n3n for n1.arrow_forwardUse mathematical induction to prove the formula for all integers n_1. 5+10+15+....+5n=5n(n+1)2arrow_forwardUse generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3arrow_forward
- 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of .arrow_forwardTower of Hanoi The result in Exercise 39 suggest that the minimum number of moves required to transfer n disks from one peg to another is given by the formula 2n1. Use the following outline to prove that this result is correct using mathematical induction. a Verify the formula for n=1. b Write the induction hypothesis. c How many moves are needed to transfer all but the largest of k+1 disks to another peg? d How many moves are needed to transfer the largest disk to an empty peg? e How many moves are needed to transfer the first k disks back onto the largest one? f How many moves are needed to accomplish steps c, d, and e? g Show that part f can be written in the form 2(k+1)1. h Write the conclusion of the proof.arrow_forwardIn Exercises 93-96, prove the property for all integers r and n, where 0rn. The sum of the numbers in the nth row of Pascal’s Triangle is 2n.arrow_forward
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