Deck 5: Sequences, Mathematical Induction, and Recursion

Question 1

Answer 1

Question 2

Transform the following summation by making the change of variable i = k + 1:

Answer 2

Question 3

Answer 3

Question 4

Use mathematical induction to prove that for all integers

Question 5

Question 6

Use mathematical induction to prove that for all integers

Question 7

Question 8

Use the formula

where m is an integer that is at least 1.

Question 9

Question 10

Use strong mathematical induction to prove that for all integers
either n is prime or n
is a product of prime numbers.

Question 11

Question 12

(a) Is P(0) true? Justify your answer.
(c) Finish the proof started in (b) above.

Question 13

Use summation notation to rewrite the following:

Question 14

Question 15

Question 16

Question 17

Use a summation symbol to rewrite the following:

Question 18

Use repeated division by 2 to find the binary representation of the number 1032. Show your
work.

Question 19

Transform the following summation by making the change of variable j = k + 1 :

Question 20

Use strong mathematical induction to prove that for all integers

Question 21

Question 22

In a Triple Tower of Hanoi, there are three poles in a row and 3n disks, three of each of n
different sizes, where n is any positive integer. Initially, one of the poles contains all the disks
placed on top of each other in triples of decreasing size. Disks are transferred one by one from
one pole to another, but at no time may a larger disk be placed on top of a smaller disk.
However, a disk may be placed on top of one of the same size. Let
be the minimum number
of moves needed to transfer a tower of 3n disks from one pole to another. Find a recurrence
relation for
Justify your answer carefully.

Question 23

Question 24

Question 25

A sequence is defined recursively as follows:
It is proposed that an explicit formula for this sequence is
Use mathematical induction to check whether this proposed formula is correct.

Question 26

Question 27

A sequence
... is defined recursively as follows:
Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers

Question 28

If appropriate, simplify your answer using one of the following reference formulas:

Question 29

The following while loop is annotated with a pre- and post-condition and also a loop invariant.
Use the loop invariant theorem to prove the correctness of the loop with respect to the pre-
and post-conditions.

Question 30

Question 31

Question 32

Question 33

In a Double Tower of Hanoi with Adjacency Requirement there are three poles in a row and
2n disks, two of each of n different sizes, where n is any positive integer. Initially pole A (at
one end of the row) contains all the disks, placed on top of each other in pairs of decreasing
size. Disks may only be transferred one-by-one from one pole to an adjacent pole and at no
time may a larger disk be placed on top of a smaller one. However a disk may be placed on
top of another one of the same size. Let C be the pole at the other end of the row and let

Question 34

Question 35

Question 36

A sequence is defined recursively as follows:
Use mathematical induction to verify that this sequence satisfies the explicit formula

Question 37