Question

1. Let n be an odd positive integer. Consider a list of n consecutive integers.

Show that the average is the middle number (that is the number in the

middle of the list when they are arranged in an increasing order). What

is the average when n is an even positive integer instead?

2.

Let x1,x2,...,xn be a list of numbers, and let ¯ x be the average of the list.Which of the following

statements must be true? There might be more than one such statement, or one, or none;

a) At least half of the numbers on the list must be bigger than ¯ x.

b) Half of the numbers on the list must be bigger than ¯ x.

c) Some of the numbers on the list must be bigger than ¯ x.

d) Not all of the numbers on the list can be bigger than ¯ x.

Answer #1

**1)** Let n be an odd positive integer

Consider a list of n consecutive integers.

let n=3, then 3 consecutive numbers (increasing order) will be a-1, a, a+1

similarly if n=5, then 5 consecutive numbers (increasing order) will be a-2, a-1, a, a+1, a+2

and so on...

so we can see that when n=3,

Average= (a-1+a+a+1)/3= a i,e the middle number

Similarly for n= 5, Average= (a-2+a-1+a+a+1+a+2)/5= a i,e the middle number

So, when n is odd then the average of the n consecutive integers
is the middle number of the list i.e,
**(n+1)/2 ^{th}** number of the list.

When n is an even positive integer instead,

Then the average of the n consecutive integers arranged in
increasing order will be the mean of the **(n/2)th**
and (**(n/2)+1)th** number of the list.

**B)** The correct statements are

c) Some of the numbers on the list must be bigger than ¯ x.

d) Not all of the numbers on the list can be bigger than ¯ x.

When the smallest of three consecutive odd integers is added to
four times the largest, it produces a result 729 more than four
times the middle integer. Find the numbers and check your
answer.

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 1.

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
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Prove that if n is a positive integer greater than 1,
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Prove that if a, b, c are integers such that a2 + b2 =
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Let n ≥ 1 be an integer. Use the Pigeonhole Principle to prove
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Let n be a positive integer and p and
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expectation below, decide whether it can be calculated with this
information, and if it can, give its value (in terms of p,
n, and r)....

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