Question

ÉTICA E SOMA DOS TERMOS
FORMULA
an = a1 + (n – 1) r
Sn=. (a1 + an)
2
1-O primeiro termo de uma PA é 100 e o trigésimo é 187. Qual a soma dos trinta primeiros termos?
2-Determine:
a) a soma dos 10 primeiros termos da PA
(2, 5, ...);
b) a soma dos 15 primeiros termos da PA (– 1, – 7, ...);
c) a soma dos 20 primeiros termos da PA (0,5; 0,75, ...).
3-O valor de x para que a seqüência (2x , x+1, 3x) seja uma PA :
4-Determinarxtalque(2x–3;2x+1;3x+1) sejam três números em P. A. nesta ordem:
5-O comandante de um destacamento militar ordenou que seus subordinados se organizassem em filas. A primeira fila era composta por 14 soldados, a segunda por 18 soldados, a terceira por 22 soldados, e assim sucessivamente. Sabe-se que o número de soldados deste destacamento é igual a 1550. Dessa forma, é correto afirmar que serão formadas:
a) 18 filas b) 20 filas c) 23 filas d) 25 filas e) 30 filas

Answer 1

1.

$a_1=100,\ a_{30}=187,\ n=30\\\\ S_{30}=\dfrac{30}{2}(100+187)\ \therefore\ \boxed{S_{30}=4305}$

2.

a)

$a_1=2,\ a_2=5,\ n=10\\\\ a_2=a_1+r\ \therefore\ 5=2+r\ \therefore\ r=3\\\\ a_{10}=a_1+9r\ \therefore\ a_{10}=2+9(3)\ \therefore\ a_{10}=29\\\\ S_{10}=\dfrac{10}{2}(2+29)\ \therefore\ \boxed{S_{10}=155}$

b)

$a_1=-1,\ a_2=-7,\ n=15\\\\ a_2=a_1+r\ \therefore\ -7=-1+r\ \therefore\ r=-6\\\\ a_{15}=a_1+14r\ \therefore\ a_{15}=-1+14(-6)\ \therefore\ a_{15}=-85\\\\ S_{15}=\dfrac{15}{2}(-1-85)\ \therefore\ \boxed{S_{15}=-645}$

c)

$a_1=\dfrac{1}{2},\ a_2=\dfrac{3}{4},\ n=20\\\\ a_2=a_1+r\ \therefore\ \dfrac{3}{4}=\dfrac{1}{2}+r\ \therefore\ r=\dfrac{1}{4}\\\\ a_{20}=a_1+19r\ \therefore\ a_{20}=\dfrac{1}{2}+19\bigg(\dfrac{1}{4}\bigg)\ \therefore\ a_{20}=\dfrac{21}{4}\\\\ S_{20}=\dfrac{20}{2}\bigg(\dfrac{1}{2}+\dfrac{21}{4}\bigg)\ \therefore\ \boxed{S_{20}=\dfrac{115}{2}}$

3.

$a_1=2x,\ a_2=x+1,\ a_3=3x\\\\ a_2=a_1+r\ \therefore\ x+1=2x+r\ \therefore\ x+r=1\\\\ a_3=a_2+r\ \therefore\ 3x=x+1+r\ \therefore\ 2x-r=1\\\\ x+r=2x-r\ \therefore\ 2r=x\ \therefore\ r=\dfrac{x}{2}\\\\ x+r=1\ \therefore\ x+\dfrac{x}{2}=1\ \therefore\ \boxed{x=\dfrac{2}{3}}$

4.

$a_1=2x-3,\ a_2=2x+1,\ a_3=3x+1\\\\ a_2=a_1+r\ \therefore\ 2x+1=2x-3+r\ \therefore\ r=4\\\\ a_3=a_2+r\ \therefore\ 3x+1=2x+1+4\ \therefore\ \boxed{x=4}$

5.

$a_1=14,\ a_2=18,\ a_3=22,\ S_n=1550\\\\ a_2=a_1+r\ \therefore\ 18=14+r\ \therefore\ r=4\\\\ a_n=a_1+(n-1)r\ \therefore\ a_n=14+(n-1)(4)\ \therefore\\\\ a_n=14+4n-4\ \therefore\ a_n=10+4n\\\\ S_n=\dfrac{n}{2}(a_1+a_n)\ \therefore\ 1550=\dfrac{n}{2}(14+10+4n)\ \therefore\\\\ 775=n(6+n)\ \therefore\ 775=6n+n^2\ \therefore\ n^2+6n-775=0\\\\ n=\dfrac{-6\pm\sqrt{6^2-4(1)(-775)}}{2(1)}=\dfrac{-6\pm\sqrt{3136}}{2}=-3\pm28\\\\ n>0\ \therefore\ n=-3+28\ \therefore\ \boxed{n=25\ \text{filas}}$