15.5 Algebraic proof — Practise

1. Prove that the sum of three consecutive integers is divisible by $3$.

Answer: $n+(n+1)+(n+2)=3(n+1)$

2. Prove that the sum of two consecutive integers is odd.

Answer: $n+(n+1)=2n+1$

3. Prove that the difference between the squares of consecutive integers is odd.

Answer: $(n+1)^2-n^2=2n+1$

4. Show that the square of an even integer is divisible by $4$.

Answer: $(2n)^2=4n^2$

5. Show that the square of an odd integer is odd.

Answer: $(2n+1)^2=2(2n^2+2n)+1$

6. Prove that the sum of two even integers is even.

Answer: $2m+2n=2(m+n)$

7. Show that the sum of three consecutive multiples of $3$ is divisible by $9$.

Answer: $3n+3(n+1)+3(n+2)=9(n+1)$

8. Show that the difference between two consecutive multiples of $5$ is divisible by $5$.

Answer: $5(n+1)-5n=5$

9. Show that $(x+2)^2-(x-2)^2$ is equivalent to $8x$.

Answer: $(x+2)^2-(x-2)^2=8x$

10. Show that the sum of three consecutive even integers is divisible by $6$.

Answer: $2n+2(n+1)+2(n+2)=6(n+1)$

11. Method choice: expand or factorise

Answer: Factorising is quicker: $(x+3)^2-(x+1)^2=((x+3)-(x+1))((x+3)+(x+1))=2(2x+4)=4x+8$.