Problems

Simplify \(\dfrac{2x+5}{6x+7} - \dfrac{2x-1}{6x+5} - \dfrac{32x+33}{36(x+1)^2-1}\). and prove that, if \(a+b+c=0\), then \(a^3+b^3+c^3=3abc\).

Solve the equations:

1. \((3x-1)^2+48x=16\),
2. \(\dfrac{35x-2}{21x+2} + \dfrac{28x-1}{21x+1} = 2\),
3. \(x^2+4y^2=2\), \\ \(8x-2y=7\).

Find a formula for the \(n\)th term of an A.P. whose first term is \(a\) and common difference \(d\), and determine the sum of the series \(1+3+5+7+\dots\) to \(n\) terms.