Question

1-Determine o número X de modo que a sequência (2x+6, 2x-2, 2-4x) seja uma P.A2-Obtenha X para que a sequência (4x-4, 6x-2, 4x+12) seja uma P.A 3- Calcular a soma dos trintas primeiros termos do P.A (4,9,14,19)4-Calcular a soma do 49° termo do P.A (5,11,17,23,....)

Answer 1

$\mathbf{1)}\ \textrm{P.A.:}\ \mathrm{(2x+6,2x-2,2-4x)}\\\\ \mathrm{a_2=a_1+r\ \to\ r=2x-2-(2x+6)\ \to\ r=-2-6\ \to\ \boxed{\mathrm{r=-8}}}\\\\ \mathrm{a_3=a_2+r\ \to\ r=2-4x-(2x-2)\ \to\ -8=4-6x\ \to\ \boxed{\mathrm{x=2}}}\\\\ \boxed{\boxed{\textrm{P.A.:}\ \mathrm{(10,2,-6)}}}$

$\mathbf{2)}\ \textrm{P.A.:}\ \mathrm{(4x-4,6x-2,4x+12)}\\\\ \mathrm{a_3=a_1+2r\ \to\ 2r=4x+12-(4x-4)\ \to\ 2r=12+4\ \to\ \boxed{\mathrm{r=8}}}\\\\ \mathrm{a_2=a_1+r\ \to\ r=6x-2-(4x-4)\ \to\ 8=2x+2\ \to\ \boxed{\mathrm{x=3}}}\\\\ \boxed{\boxed{\textrm{P.A.:}\ \mathrm{(8,16,24)}}}$

$\mathbf{3)}\ \textrm{P.A.:}\ \mathrm{(4,9,14,19,\dots)}\\\\ \mathrm{a_2=a_1+r\ \to\ r=9-4\ \to\ \boxed{\mathrm{r=5}}}\\\\ \mathrm{a_{30}=a_1+29r\ \to\ a_{30}}=4+29(5)\ \to\ a_{30}=4+145\ \to\ \boxed{\mathrm{a_{30}=149}}}\\\\ \mathrm{S_n=\dfrac{(a_1+a_n)n}{2}\ \to\ S_{30}=\dfrac{(a_1+a_{30})30}{2}\ \to\ S_{30}=15(4+149)\ \to}\\\\ \mathrm{\to\ S_{30}=15(153)\ \to\ \boxed{\boxed{\mathrm{S_{30}=2295}}}}$

$\mathbf{4)}\ \textrm{P.A.:}\ \mathrm{(5,11,17,23,\dots)}\\\\ \mathrm{a_2=a_1+r\ \to\ r=11-5\ \to\ \boxed{\mathrm{r=6}}}\\\\ \mathrm{a_{49}=a_1+48r\ \to\ a_{49}=5+48(6)\ \to\ a_{49}=5+288\ \to\ \boxed{\boxed{\mathrm{a_{49}=293}}}}$