Question Type 1: Simple equation proofs Exercises

Show that $\displaystyle \frac{\frac1a + \frac1b}{\frac1b - \frac1a} = \frac{a+b}{a-b}$, for $a,b\neq 0$ and $a\neq b$.

Show that $\displaystyle \frac{1}{x+h} - \frac{1}{x} = -\frac{h}{x(x+h)}$, for $x \neq 0$ and $x+h \neq 0$.

Show that $\left(1 + \frac{1}{x}\right)^2 - \left(1 - \frac{1}{x}\right)^2 = \frac{4}{x}$, for $x \neq 0$.

Prove that $\sqrt{a} - \sqrt{b} = \dfrac{a - b}{\sqrt{a} + \sqrt{b}}$, for distinct positive real numbers $a, b$.

Prove that $\displaystyle \frac{x^2 + x + 1}{x+2} = x-1 + \frac{3}{x+2}$, for $x\neq -2$.

Prove that $\displaystyle \frac{1}{x - 1} - \frac{1}{x + 1} = \frac{2}{x^2 - 1}$, for $x \neq \pm1$.

Verify that $\displaystyle \frac{1+x}{1-x} - \frac{1-x}{1+x} = \frac{4x}{1-x^2}$, for $x\neq \pm1$.

Show that $\displaystyle \frac{1}{\sqrt{x}+\sqrt{y}} + \frac{1}{\sqrt{x}-\sqrt{y}} = \frac{2\sqrt{x}}{x-y}$, for $x\neq y$, $x,y>0$.

Prove that $\frac{x^3 - y^3}{x - y} = x^2 + xy + y^2$, for $x \neq y$.

Prove that $\dfrac{(n+1)!}{n!} = n+1$, for integer $n \geq 0$.

Prove that $\frac{1}{m+1} + \frac{1}{m^2 + m} = \frac{1}{m}$, where $m \neq 0,-1$.

Show that $\frac{a^2 - b^2}{a - b} = a + b$, for $a \neq b$.