mathematics, algebraic ratio and proportion
Question: If $a,b,c$ are be continued proportion, prove that $(i).\dfrac{a^3-b^3}{a+b+c}=a(a-b)$ $(ii).\dfrac{a^3+b^3}{a-b+c}=a(a+b)$ $(iii).\dfrac{a^3+b^3}{b^3+c^3}=\dfrac{a(a+b)}{c(b+c)}$ $(iv).\dfrac{a^3-b^3}{b^3-c^3}=\dfrac{a(a-b)}{c(b-c)}$ Question: $\dfrac{x^{3}+3 x}{x^{2}+1}=\dfrac{14}{13} $ Solution: $\dfrac{x^{3}+3 x}{x^{2}+1}=\dfrac{14}{13} $ $\Rightarrow \dfrac{x^{3}+3 x+3 x^{2}+1}{x^{3}+3 x-3 x^{2}-1}=\dfrac{14+13}{14-13}$ $\Rightarrow \dfrac{(x+1)^{3}}{(x-1)^{3}}=\dfrac{27}{1}$ $\Rightarrow\left(\dfrac{x+1}{x-1}\right)^{3}=27$ $\Rightarrow \dfrac{x+1}{x-1}=3$ $\Rightarrow 3x-3=x+1$ $\Rightarrow 2x=4$ $\therefore x=2$ Questions: 1. If $\dfrac{x+y}{3 a-b}=\dfrac{y+z}{3 b-c}=\dfrac{z+x}{3 c-a}$ show that $\dfrac{x+y+z}{a+b+c}=\dfrac{a x+b y+c z}{a^{2}+b^{2}+c^{2}}$ 2. If $\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}$ , show that $(i)$ $\dfrac{x^{2}-y z}{a^{2}-b c}=\dfrac{y^{2}-z x}{b^{2}-c a}=\dfrac{z^{2}-x y}{c^{2}-a b}$ $(ii)$ $\dfrac{x^3}...