Question

afla numarul natural x care verifica egalitatea x^2 -1 =2(1+3+3^2+........+3^101)

Answer 1

$1+3+3^2+\dots +3^{101}$ este suma unei progresii geometrice cu ratia 3. Deci, $1+3+3^2+\dots +3^{101} =1 \cdot \dfrac{3^{102}-1}{3-1}= \dfrac{3^{102}-1}{2}$ .
$x^2-1 = 2 \cdot \dfrac{3^{102}-1}{2} \iff x^2-1 = 3^{102}-1$ adica $x^2=3^{102} \implies x = 3^{51}$