Question

Sa se arate ca daca:
1) nr. a patrat, b patrat, c patrat sunt in progresie aritmetrica,atunci si 1`\b+c, 1\c+a, 1\a+b sunt in progresie aritmetica.

Answer 1

$Pentru~a~demonstra~ca~numerele~ \frac{1}{b+c},~ \frac{1}{c+a}~si~ \frac{1}{a+b}~sunt~in \\ \\ progresie~aritmetica~trebuie~sa~demonstram~ca~ \frac{1}{c+a}~este~media \\ \\ aritmetica~dintre~ \frac{1}{b+c}~si~ \frac{1}{a+b},~adica: \\ \\ \displaystyle \frac{1}{c+a}= \frac{ \frac{1}{b+c}+ \frac{1}{a+b} }{2} \Leftrightarrow \frac{2}{c+a}= \frac{a+2b+c}{(a+b)(b+c)} \Leftrightarrow \\ \\ \Leftrightarrow 2ab+2ac+2b^2+2bc=ac+2bc+c^2+a^2+2ab+ac \Leftrightarrow$

$\Displaystyle \Leftrightarrow 2b^2=a^2+c^2.~(*) \\ \\ Dar~a^2,~b^2,~c^2~sunt~in~progresie~aritmetica.~Rezulta~ca~relatia~care \\ \\ trebuie~demonstrata~(*)~ este ~adevarata.\\ \\ \\ l$