Question

Rezolvati in Q ecuatia:
X(1+1 pe 2+1 pe 3+...+1 pe 100)=100- 1 pe 2-2 pe 3-3 pe 4-...-99 pe 100

Multumesc anticipat!

Answer 1

Tot jmenul consta in descompunerea lui 100 ca suma de 1 ( 100=1+1+1+....+1)

$x\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+ \dfrac{1}{100}\right)=100-\dfrac{1}{2}-\dfrac{2}{3}-\ldots -\dfrac{99}{100}\\x\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+ \dfrac{1}{100}\right)=1+\left(1-\dfrac{1}{2}\right)+\left(1-\dfrac{2}{3}\right)+\ldots +\left(1-\dfrac{99}{100}\right)\\x\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+ \dfrac{1}{100}\right)=1+ \dfrac{2-1}{2}+\dfrac{3-2}{3}+\ldots +\dfrac{100-99}{100}$

$x\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+ \dfrac{1}{100}\right)=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots +\dfrac{1}{100}\\\boxed{\boxed{\boxed{\bold{x=1}}}}$