Question

a p.a (-20,-14,-8....,x) tem a soma do seus termos igual a 946 . calcule x?

Answer 1

$\mathrm{PA=(-20,-14,-8,\dots,x)\ \ \| \ \ S_n=946\ \ \| \ \ r=6}\\\\ \textbf{Pelo termo geral de uma PA, vem:}\\\\ \mathrm{a_n=a_1+(n-1).r\ \to\ x=-20+(n-1).6}\\ \mathrm{x=-20+6n-6\ \to\ x=6n-26\ \mathbf{(I)}}\\\\ \textbf{Pela soma dos termos de uma PA, vem:}\\\\ \mathrm{S_n=\dfrac{(a_1+a_n).n}{2}\ \to\ 946=\dfrac{(-20+6n-26).n}{2}}\\\\ \mathrm{2.946=(6n-46).n\ \to\ 946=(3n-23).n}\\ \mathrm{946=3n^2-23n\ \to\ 0=3n^2-23n-946}$

$\textbf{Atrav\'es da f\'ormula quadr\'atica, teremos que:}\\\\ \mathrm{n=\dfrac{-(-23)\pm\sqrt{(-23)^2-4.3.(-946)}}{2.3}=}\\\\ \mathrm{=\dfrac{23\pm\sqrt{529+11352}}{6}=\dfrac{23\pm\sqrt{11881}}{6}=}\\\\ \mathrm{=\dfrac{23\pm109}{6}\ \to\ n\ \textgreater \ 0\ \to\ n=\dfrac{23+109}{6}=}\\\\ \mathrm{=\dfrac{132}{6}=22\ termos}\\\\ \textbf{Substituindo o valor de n em I:}\\\\ \mathrm{x=6n-26\ \to\ x=6.22-26}\\ \mathrm{x=132-26\ \to\ \mathbf{x=106}}$