Proof & Reasoning (DP IB Analysis & Approaches (AA): SL): Exam Questions

Show that $\frac{1}{n+1}+\frac{1}{n^2+n}=\frac{1}{n}.$

For$f\left(x\right)=x^2-10x+17$, prove that$f\left(x\right)\ge -8$ for all values of $x.$

Prove that the exterior angle in any triangle is equal to the sum of the two opposite interior angles. You may use the diagram below to help.

Consider the function$f\left(x\right)=5x^2+4x+1.$ Show that$f\left(x\right)$ is positive for all values of $x.$

Consider two consecutive positive integers, $n$ and $n+1$.

Show that the difference of their squares is equal to the sum of the two integers.

Prove that $(2q-1)(q-3)-3(q-4{)}^2=-q^2+17q-45.$

Prove that the square of an odd number is always odd.

Prove that the sum of the squares of any two consecutive odd integers is even.

Prove that the sum of any three consecutive even numbers is a multiple of 6.

(i) Prove that

$\frac{a}{\left({\displaystyle \frac{b}{c}}\right)}=\frac{ac}{b}$

(ii) Specify any cases for which the relation in part (a)(i) is not valid.

Prove that $(p-q{)}^2=(q-p{)}^2$ for all numbers$p$ and $q.$

Prove that the product of two odd numbers is odd.

The sum of squares of two consecutive integers is 313.  Find the possible values of the integers.

Prove that the sum of the cubes of any two consecutive odd integers is divisible by four.

Prove that $\frac{a^2-a-6}{a+4}\times \frac{a^2-16}{a^2+2a}=a-7+\frac{12}{a}.$

State any values of $a$ for which this mathematical statement does not hold true.

Prove that there are no integers$p$ and $q$ that satisfy the equation

$4p^2-q^2=49$

Prove the binomial coefficient identity 

$\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}n-1\\ k\end{array}\right)+\left(\begin{array}{c}n-1\\ k-1\end{array}\right).$

Prove that the sum of all integers between 600 and 1400 (inclusive) that are not divisible by 7 is equal to $685885.$