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4. A wheel of fortune has the integers from 1 to 25 placed on it

in a random manner. Show that regardless of how the numbers

are positioned on the wheel, there are three adjacent numbers

whose sum is at least 39.

Work:

Let suppose that there are not three adjacent numbers whose sum is at least 39 then for every set of 3 adjacent numbers their sum is less than 39

Since all the numbers are integers for every set of 3 adjacent numbers their sum is less than or equal to 38

Let select the 24 numbers around the “1”, from these 24 numbers we can create 8 sets of 3 consecutive adjacent numbers , then the total sum is less than or equal to 8(38)+1 = 305

So we have that the sum of 1+2+3+……+25 305 (but this is false) because

1+2+….25 = 25(26)/2 = 325 > 305

Then we proved that regardless of how the numbers are positioned on the wheel, there are three adjacent numbers whose sum is at least 39.

7. A lumberjack has 4n + 110 logs in a pile consisting of n layers.

Each layer has two more logs than the layer directly above

it. If the top layer has six logs, how many layers are there?

Work:

The 1st layer (the top one) has 6 logs

The 2nd layer has 6+1(2) logs

The 3rd layer has 6+2(2)

The 4th layer has 6+3(2)

…………….

The nth layer 6 +(n-1)(2)

Total number of layers is: 6n +2(1+2+3+….+n-1) = 6n + 2(n-1)n/2 =

6n+n(n-1) = (n+5)n

Then: n(n+5)=4n+110 n2+5n = 4n +110 n2 + n-110 =0 n = [-1441]/2 = [-121]/2 n = (-1+21)/2 = 10 (Taking the positive solution)

Answer: 10 layers

Exercise 4.2:

16. Give a recursive definition for the set of all

a) positive even integers

b) nonnegative even integers

Answers:

a) Let S the set of positive even integers

Recursive definition of S:

i) 2 S ii) If x S x+2 S

b)…...

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