Quiz Results

Quiz	Proof
Name	D a n i e l G a l l a c h e r
Email	d a n i e l _ g a l l a c h e r @ o u t l o o k . c o m
Result	PASSED
Score	12 / 14 (85.7%)
Passing score	11.2
Quiz took	33 min 02 sec
Quiz finished at	2022-08-05 12:26:57

The expression 2n represents

Awarded Points: 1

Correct Response: an even number

User Response: an even number

The expression 2n+1 represents

Awarded Points: 1

Correct Response: an odd number

User Response: an odd number

Which expressions represent three consecutive integers?

Awarded Points: 1

Correct Response: n, n+1, n+2

User Response: n, n+1, n+2

Which expressions represent three consecutive even integers?

Awarded Points: 1

Correct Response: 2n, 2n+2, 2n+4

User Response: 2n, 2n+2, 2n+4

Which expressions represent three consecutive odd integers?

Awarded Points: 1

Correct Response: 2n+1, 2n+3, 2n+5

User Response: 2n+1, 2n+3, 2n+5

What do the following lines of algebra prove? 2n + \left(2n+2\right) + \left(2n+4\right)\phantom{\rule{0ex}{0ex}}= 6n+6\phantom{\rule{0ex}{0ex}}= 6\left(n+1\right)

Awarded Points: 1

Correct Response: The sum of three consecutive even numbers is always a multiple of 6

User Response: The sum of three consecutive even numbers is always a multiple of 6

What do the following lines of algebra prove? \left(2n+1\right) + \left(2n+3\right) + \left(2n+5\right)\phantom{\rule{0ex}{0ex}}= 6n+9\phantom{\rule{0ex}{0ex}}= 3\left(2n+3\right)

Awarded Points: 1

Correct Response: The sum of three consecutive odd numbers is always a multiple of 3

User Response: The sum of three consecutive odd numbers is always a multiple of 3

What do the following lines of algebra prove? \left(2n{\right)}^{2} + \left(2m{\right)}^{2}\phantom{\rule{0ex}{0ex}}= 4{n}^{2} + 4{m}^{2}\phantom{\rule{0ex}{0ex}}= 4\left({n}^{2} + {m}^{2}\right)

Awarded Points: 1

Correct Response: The sum of the squares of any two even positive integers is always a multiple of 4

User Response: The sum of the squares of any two even positive integers is always a multiple of 4

What do the following lines of algebra prove? \left(2n+1{\right)}^{2} + \left(2m+1{\right)}^{2}\phantom{\rule{0ex}{0ex}}= \left(2n+1\right)\left(2n+1\right) + \left(2m+1\right)\left(2m+1\right)\phantom{\rule{0ex}{0ex}}= \left(4{n}^{2} +2n+2n+1\right) + \left(4{m}^{2} +2m+2m+1\right)\phantom{\rule{0ex}{0ex}}= \left(4{n}^{2} 4n+1\right) + \left(4{m}^{2} +4m+1\right)\phantom{\rule{0ex}{0ex}}= 4{n}^{2}+4n+4{m}^{2}+4m+2\phantom{\rule{0ex}{0ex}}= 2\left(2{n}^{2}+2n+2{m}^{2}+2m+1\right)

Awarded Points: 1

Correct Response: The sum of the squares of any two odd positive integers is always even

User Response: The sum of the squares of any two odd positive integers is always even

What do the following lines of algebra prove? {n}^{2} + \left(n+1{\right)}^{2}\phantom{\rule{0ex}{0ex}}= {n}^{2} + \left(n+1\right)\left(n+1\right)\phantom{\rule{0ex}{0ex}}= {n}^{2 }+ {n}^{2} + n + n + 1\phantom{\rule{0ex}{0ex}}= 2{n}^{2} + 2n + 1\phantom{\rule{0ex}{0ex}}= 2\left({n}^{2}+n\right) + 1

Awarded Points: 1

Correct Response: The sum of the squares of any two consecutive integers is always an odd number

User Response: The sum of the squares of any two consecutive integers is always an odd number

What do the following lines of algebra prove? {\left(2n\right)}^{2} + \left(2n+2{\right)}^{2}\phantom{\rule{0ex}{0ex}}= \left(2n{\right)}^{2}+\left(2n+2\right)\left(2n+2\right)\phantom{\rule{0ex}{0ex}}= 4{n}^{2 }+ \left(4{n}^{2} + 4n + 4n + 4\right)\phantom{\rule{0ex}{0ex}}= 4{n}^{2 }+ \left(4{n}^{2} + 8n + 4\right)\phantom{\rule{0ex}{0ex}}= 8{n}^{2} + 8n + 4\phantom{\rule{0ex}{0ex}}= 8\left({n}^{2}+n\right) + 4

Awarded Points: 1

Correct Response: The sum of the squares of any two consecutive even numbers is always 4 more than a multiple of 8

User Response: The sum of the squares of any two consecutive even numbers is always 4 more than a multiple of 8

What do the following lines of algebra prove? {\left(2n+1\right)}^{2} + \left(2n+3{\right)}^{2}\phantom{\rule{0ex}{0ex}}= \left(2n+1\right)\left(2n+1\right)+\left(2n+3\right)\left(2n+3\right)\phantom{\rule{0ex}{0ex}}= \left(4{n}^{2 }+ 2n + 2n + 1\right) + \left(4{n}^{2} + 6n + 6n + 9\right)\phantom{\rule{0ex}{0ex}}= \left(4{n}^{2 }+ 4n + 1\right) + \left(4{n}^{2} + 12n + 9\right)\phantom{\rule{0ex}{0ex}}= 8{n}^{2} + 16n + 10\phantom{\rule{0ex}{0ex}}= 8{n}^{2} + 16n + 8 + 2\phantom{\rule{0ex}{0ex}}= 8\left({n}^{2}+2n +1\right) + 2

Awarded Points: 1

Correct Response: The sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8

User Response: The sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8

What do the following lines of algebra prove about the values (n+1) and n? {\left(n+1\right)}^{2} - {n}^{2}\phantom{\rule{0ex}{0ex}}= \left(n+1\right)\left(n+1\right) - {n}^{2}\phantom{\rule{0ex}{0ex}}= \left({n}^{2 }+ n + n + 1\right) - {n}^{2}\phantom{\rule{0ex}{0ex}}= \left({n}^{2 }+ 2n + 1\right) - {n}^{2}\phantom{\rule{0ex}{0ex}}= 2n+1

Awarded Points: 0

Correct Response: The difference between the squares of any two consecutive integers is equal to the sum of these integers

User Response: The difference between any two odd consecutive integers squared is odd

It is possible to write out a proof to show that the sum of (n+2)(n+1) and (n+2) is always _____________________

Responses

Awarded Points: 1

Correct Response: an even number

User Response: an even number

Awarded Points: 1

Correct Response: an odd number

User Response: an odd number

Awarded Points: 1

Correct Response: n, n+1, n+2

User Response: n, n+1, n+2

Awarded Points: 1

Correct Response: 2n, 2n+2, 2n+4

User Response: 2n, 2n+2, 2n+4

Awarded Points: 1

Correct Response: 2n+1, 2n+3, 2n+5

User Response: 2n+1, 2n+3, 2n+5

Awarded Points: 1

Correct Response: The sum of three consecutive even numbers is always a multiple of 6

User Response: The sum of three consecutive even numbers is always a multiple of 6

Awarded Points: 1

Correct Response: The sum of three consecutive odd numbers is always a multiple of 3

User Response: The sum of three consecutive odd numbers is always a multiple of 3

Awarded Points: 1

Correct Response: The sum of the squares of any two even positive integers is always a multiple of 4

User Response: The sum of the squares of any two even positive integers is always a multiple of 4

Awarded Points: 1

Correct Response: The sum of the squares of any two odd positive integers is always even

User Response: The sum of the squares of any two odd positive integers is always even

Awarded Points: 1

Correct Response: The sum of the squares of any two consecutive integers is always an odd number

User Response: The sum of the squares of any two consecutive integers is always an odd number

Awarded Points: 1

Correct Response: The sum of the squares of any two consecutive even numbers is always 4 more than a multiple of 8

User Response: The sum of the squares of any two consecutive even numbers is always 4 more than a multiple of 8

Awarded Points: 1

Correct Response: The sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8

User Response: The sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8

Awarded Points: 0

Correct Response: The difference between the squares of any two consecutive integers is equal to the sum of these integers

User Response: The difference between any two odd consecutive integers squared is odd

Awarded Points: 0

Correct Response: a square number

User Response: an even number