Download Worksheet: Year 9 - Proofs Homework

Year 9 Proofs Homework
1. For each of these statements, provide a counterexample to show that the statement is false.
a. All numbers of the form 2𝑛 − 1 where 𝑛 > 1 and 𝑛 is an integer, are prime.
b. The square of a number is greater than the original number.
c. Four infinitely long straight lines intersect at 6 points.
2. The following sequences use a rule to generate the next number in the sequence. Find a rule
which gives the final term instantly from the initial underlined terms (you can call these 𝑎,
and where relevant, 𝑏 as well).
a. 3, 4, 7, 11, 18
b. 2, 3, 5, 9, 17, 33, 65
c. 1, 1, 3, 7, 17, 41, 99
3. Divisibility proofs:
a. Prove that the sum of five consecutive numbers is divisible by 5.
b. Prove that the sum of three consecutive odd numbers gives a remainder of 3 when
divided by 6.
c. How many consecutive integers would we have to add together to guarantee it is
divisible by 4?
 Prove that there is exactly one sequence of five consecutive positive integers in which the
sum of the squares of the first three integers is equal to the sum of the squares of the other
two integers. [Source: IMO]
Year 9 Proofs Homework
1. For each of these statements, provide a counterexample to show that the statement is false.
a. All numbers of the form 2𝑛 − 1 where 𝑛 > 1 and 𝑛 is an integer, are prime.
b. The square of a number is greater than the original number.
c. Four infinitely long straight lines intersect at 6 points.
2. The following sequences use a rule to generate the next number in the sequence. Find a rule
which gives the final term instantly from the initial underlined terms (you can call these 𝑎,
and where relevant, 𝑏 as well).
a. 3, 4, 7, 11, 18
b. 2, 3, 5, 9, 17, 33, 65
c. 1, 1, 3, 7, 17, 41, 99
3. Divisibility proofs:
a. Prove that the sum of five consecutive numbers is divisible by 5.
b. Prove that the sum of three consecutive odd numbers gives a remainder of 3 when
divided by 6.
c. How many consecutive integers would we have to add together to guarantee it is
divisible by 4?
 Prove that there is exactly one sequence of five consecutive positive integers in which the
sum of the squares of the first three integers is equal to the sum of the squares of the other
two integers. [Source: IMO]
www.drfrostmaths.com
Answers:
1) a) 15 is not prime, or 63
b) Any number between 0 and 1 (exclusive) c) If any of the
lines are parallel or more than two lines intersect at a single point.
2) –
a. 2𝑎 + 3𝑏
b. 64𝑎 − 63
c. 29𝑎 + 70𝑏
3) a) 𝑛 + (𝑛 + 1) + (𝑛 + 2) + (𝑛 + 3) + (𝑛 + 4) = 5𝑛 + 10 = 5(𝑛 + 2)
b) (2𝑛 + 1) + (2𝑛 + 3) + (2𝑛 + 5) = 6𝑛 + 9 = 6(𝑛 + 1) + 3 which is 3 more than a
multiple of 6.
c) 8 consecutive numbers required. 𝑛 + (𝑛 + 1) + ⋯ + (𝑛 + 7) = 8𝑛 + 28 = 4(2𝑛 + 7)
4) 𝑛2 + (𝑛 + 1)2 + (𝑛 + 2)2 = (𝑛 + 3)2 + (𝑛 + 4)2
Solutions (after expanding and solving) are 𝑛 = −2 𝑜𝑟 𝑛 = 10, but 𝑛 must be positive, so
𝑛 = 10.
www.drfrostmaths.com