igureMaths
Log in
Start free
←
Proof
Get dealt questions instead
→
Browse Proof
65 questions at your level
Difficulty
All
Level 2
Level 3
Level 4
Level 5
Proof by deduction
30 questions
Lesson
Not started
Mark as done
Given that
x
x
x
is a real number such that
x
3
+
x
=
12
x^3 + x = 12
x
3
+
x
=
12
(a)
use proof by contradiction to prove that
x
<
3
x < 3
x
<
3
(4 marks)
(b)
Show that
2
k
2
−
12
k
+
21
2k^2 - 12k + 21
2
k
2
−
12
k
+
21
is positive for all real values of
k
k
k
(2 marks)
●●●●●
Level 5
6 marks
Start
→
Mark as done
(a)
Prove, using algebra, that
5
n
+
1
−
5
n
5^{n+1} - 5^n
5
n
+
1
−
5
n
is a multiple of 20 for all positive integers
n
n
n
(3 marks)
(b)
A student was asked to use proof by contradiction to prove the following statement.
“For all integers
n
n
n
, if
n
2
n^2
n
2
is a multiple of 4 then
n
n
n
is even”
The start of the student's proof is shown below.
Assume that
n
2
n^2
n
2
is not a multiple of 4 and that
n
n
n
is odd.
⇒
n
=
2
k
+
1
\Rightarrow n = 2k + 1
⇒
n
=
2
k
+
1
for some integer
k
k
k
⇒
n
2
=
4
k
2
+
4
k
+
1
=
4
(
k
2
+
k
)
+
1
\Rightarrow n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1
⇒
n
2
=
4
k
2
+
4
k
+
1
=
4
(
k
2
+
k
)
+
1
Explain why the student's proof cannot be completed.
(1 mark)
(c)
Use proof by contradiction to prove the statement.
(3 marks)
●●●●●
Level 5
7 marks
Start
→
Mark as done
Prove that the difference between the squares of any two consecutive odd numbers is a multiple of
8
8
8
.
●●●●●
Level 5
4 marks
Start
→
Mark as done
Prove that if
n
n
n
is odd then
n
2
−
1
n^2 - 1
n
2
−
1
is a multiple of
8
8
8
.
●●●●●
Level 5
3 marks
Start
→
Mark as done
(a)
In an arithmetic series, the first term is
a
a
a
and the common difference is
d
d
d
.
Prove that the sum of the first
n
n
n
terms of the series is given by
S
n
=
n
2
[
2
a
+
(
n
−
1
)
d
]
S_n = \frac{n}{2}\left[2a + (n - 1)d\right]
S
n
=
2
n
[
2
a
+
(
n
−
1
)
d
]
(3 marks)
Dev borrows £
3000
3000
3000
from a relative. He repays the loan in monthly instalments.
Dev repays £
55
55
55
in month
1
1
1
, £
65
65
65
in month
2
2
2
, £
75
75
75
in month
3
3
3
and so on, so that the monthly repayments form an arithmetic sequence.
(b)
Show that Dev repays £
145
145
145
in month
10
10
10
.
(1 mark)
Given that Dev takes exactly
n
n
n
months to repay the loan completely,
(c)
show that
n
2
+
10
n
−
600
=
0
n^2 + 10n - 600 = 0
n
2
+
10
n
−
600
=
0
(2 marks)
(d)
Solve the equation in part (c).
(2 marks)
(e)
Hence state the number of months Dev takes to repay the loan, giving a brief reason for your answer.
(1 mark)
●●●●●
Level 5
9 marks
Start
→
Proof by contradiction
31 questions
Lesson
Not started
Mark as done
Given that
x
x
x
is a real number such that
x
3
+
x
=
12
x^3 + x = 12
x
3
+
x
=
12
(a)
use proof by contradiction to prove that
x
<
3
x < 3
x
<
3
(4 marks)
(b)
Show that
2
k
2
−
12
k
+
21
2k^2 - 12k + 21
2
k
2
−
12
k
+
21
is positive for all real values of
k
k
k
(2 marks)
●●●●●
Level 5
6 marks
Start
→
Mark as done
(a)
Prove, using algebra, that
5
n
+
1
−
5
n
5^{n+1} - 5^n
5
n
+
1
−
5
n
is a multiple of 20 for all positive integers
n
n
n
(3 marks)
(b)
A student was asked to use proof by contradiction to prove the following statement.
“For all integers
n
n
n
, if
n
2
n^2
n
2
is a multiple of 4 then
n
n
n
is even”
The start of the student's proof is shown below.
Assume that
n
2
n^2
n
2
is not a multiple of 4 and that
n
n
n
is odd.
⇒
n
=
2
k
+
1
\Rightarrow n = 2k + 1
⇒
n
=
2
k
+
1
for some integer
k
k
k
⇒
n
2
=
4
k
2
+
4
k
+
1
=
4
(
k
2
+
k
)
+
1
\Rightarrow n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1
⇒
n
2
=
4
k
2
+
4
k
+
1
=
4
(
k
2
+
k
)
+
1
Explain why the student's proof cannot be completed.
(1 mark)
(c)
Use proof by contradiction to prove the statement.
(3 marks)
●●●●●
Level 5
7 marks
Start
→
Mark as done
Use proof by contradiction to prove that
3
\sqrt{3}
3
is irrational. (You may use the result that if
a
2
a^2
a
2
is a multiple of
3
3
3
then
a
a
a
is a multiple of
3
3
3
.)
●●●●●
Level 5
5 marks
Start
→
Mark as done
Use proof by contradiction to show that there is no smallest positive rational number.
●●●●●
Level 5
3 marks
Start
→
Mark as done
Use proof by contradiction to show that
log
2
3
\log_{2} 3
lo
g
2
3
is irrational. (5 marks)
●●●●●
Level 5
5 marks
Start
→