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Algebraic proof
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41 questions at your level
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Level 5
Expressing generality — the language of proof
18 questions
Lesson
Not started
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Prove that the product of any two odd numbers is odd.
●●●●●
Level 5
4 marks
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3
a
:
2
c
=
4
:
5
3a : 2c = 4 : 5
3
a
:
2
c
=
4
:
5
b
:
4
c
=
3
:
2
b : 4c = 3 : 2
b
:
4
c
=
3
:
2
Show that
a
+
b
:
b
+
c
=
14
:
15
a + b : b + c = 14 : 15
a
+
b
:
b
+
c
=
14
:
15
(3 marks)
●●●●●
Level 5
3 marks
Start
→
Mark as done
There are only green pens and black pens in a box.
The number of green pens is
3
3
3
times the number of black pens.
Zara takes at random one pen from the box, records its colour and puts it back in the box.
She does this
n
n
n
times (
n
⩾
2
n \geqslant 2
n
⩾
2
).
Write down an expression in
n
n
n
for the probability that she takes at least one green pen and at least one black pen.
(2 marks)
●●●●●
Level 5
2 marks
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→
Proving divisibility results
13 questions
Lesson
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Prove that the difference between the squares of any two consecutive even numbers is a multiple of 4.
●●●●●
Level 5
3 marks
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→
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Prove that
n
3
−
n
n^3 - n
n
3
−
n
is a multiple of 6 for every integer
n
n
n
.
●●●●●
Level 5
4 marks
Start
→
Mark as done
Prove that the sum of any four consecutive integers is
never
a multiple of 4.
●●●●●
Level 5
3 marks
Start
→
Mark as done
Prove that
n
2
+
3
n
n^2 + 3n
n
2
+
3
n
is even for every integer
n
n
n
.
●●●●●
Level 5
3 marks
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→
Mark as done
Prove that
n
3
+
3
n
2
+
2
n
n^3 + 3n^2 + 2n
n
3
+
3
n
2
+
2
n
is a multiple of 6 for every integer
n
n
n
.
●●●●●
Level 5
4 marks
Start
→
Proofs with squares
14 questions
Lesson
Not started
Mark as done
Prove that
x
2
−
6
x
+
10
x^2 - 6x + 10
x
2
−
6
x
+
10
is positive for all values of
x
x
x
.
●●●●●
Level 5
3 marks
Start
→
Mark as done
Prove that the sum of the squares of any two consecutive integers is odd.
●●●●●
Level 5
3 marks
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→
Mark as done
Priya says, "
n
2
+
n
+
11
n^2 + n + 11
n
2
+
n
+
11
is a prime number for every positive integer
n
n
n
."
Show that Priya is wrong.
●●●●●
Level 5
2 marks
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→
Mark as done
Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8.
●●●●●
Level 5
4 marks
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→
Mark as done
Ellie says, "
n
2
−
n
+
5
n^2 - n + 5
n
2
−
n
+
5
is a prime number for every positive integer
n
n
n
."
Show that Ellie is wrong.
●●●●●
Level 5
2 marks
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→
Mark as done
Prove that the sum of the squares of any two consecutive odd numbers is always
2
2
2
more than a multiple of
8
8
8
. (4 marks)
●●●●●
Level 5
4 marks
Start
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