AQA A-Level Maths: Problem Solving and Proof

Three consecutive integers may be written as $n - 1$n−1, $n$n and $n + 1$n+1, where $n$n is an integer.

(a) Show that the sum of the squares of these three consecutive integers can be written as $3n^2 + 2$3n2+2. (3 marks)

(b) Hence prove that the sum of the squares of any three consecutive integers is never divisible by $3$3. (4 marks)

(c) The sum of the squares of three consecutive integers is $110$110. Find all possible sets of three such integers. (5 marks)

(a) Prove that the difference between the squares of any two consecutive odd numbers is a multiple of $8$8. (6 marks)

(b) Aleksy makes the following conjecture:

"The difference between the squares of any two consecutive odd numbers is a multiple of $16$16."

Determine, with justification, whether Aleksy's conjecture is true or false. (4 marks)

(a) Prove that, for any integer $n$n, if $n^2$n2 is divisible by $3$3 then $n$n is divisible by $3$3. (3 marks)

(b) Hence prove by contradiction that $\sqrt{3}$3​ is irrational.

You may assume that any rational number can be written as $\dfrac{a}{b}$ba​, where $a$a and $b$b are integers with no common factor other than $1$1 and $b \neq 0$b=0. (5 marks)

Prove by exhaustion that $n^2 + n + 1$n2+n+1 is an odd number for every integer $n$n. (6 marks)

A student is investigating the expression $n^2 - n + 11$n2−n+11, where $n$n is a positive integer. She writes:

"Testing the values $n = 1, 2, 3, 4, 5$n=1,2,3,4,5 gives $11, 13, 17, 23, 31$11,13,17,23,31, which are all prime. So $n^2 - n + 11$n2−n+11 is prime for every positive integer $n$n."

Disprove the student's claim, and explain the flaw in her reasoning. (5 marks)

A fair ordinary six-sided die is rolled twice and the two scores are added together.

Show that the probability that the total is a prime number is $\dfrac{5}{12}$125​. (4 marks)