You may also like

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Agile Algebra

Observe symmetries and engage the power of substitution to solve complicated equations.

Fibonacci Fashion

Stage: 5 Challenge Level:

Given $F_n={1\over\sqrt5}(\alpha^n-\beta^n)$ where $\alpha$ and $\beta$ are solutions of the quadratic equation $x^2-x-1=0$ and $\alpha > \beta$ show that

(1)$\alpha\beta =-1$, $\alpha + \beta = 1$

(2)${ 1\over \alpha}+{1\over \alpha^2} = {1\over \beta} + {1\over \beta^2}=1$

(3)$F_1=F_2=1$ and $F_n + F_{n+1} = F_{n+2}$ and hence $F_n$ is the $n$th Fibonacci number and

(4) the sum $1 + F_1 + F_2 + \ldots F_n$ gives another Fibonacci number.