Determine a soma dos seis primeiros termos de cada PG
A) (√3, √3,√3,...)
B) (32,16,8,...)
C) (2,6,18,...)
D) (1/5, 2/15, 4/45) *Fracionário
E) (4,2/3, 1/9,...) *
F) (1/3, -1, 3,...) *
Soluções para a tarefa
a) (√3,√3,√3 ,...)
q = 1
Sn = a1 . (qⁿ - 1)/q - 1
Sn = √3 (1⁶ -1)/1-1
Sn = √3 . 0/0
Sn = 0
b) (32,16,8, ...)
q= 16/32 ∴q = 1/2
Sn = 32 . [(1/2)⁶ - 1]/1/2 - 1
Sn = 32. [1/64 - 1]/-1/2
Sn = 32. [- 63/64]/-1/2
Sn = - 2016/64/-1/2
Sn = -2016/64 . -2/1
Sn = 4032/64
Sn = 63
c) (2,6,18,...)
q= 6:2 = 3
Sn = 2. (3⁶ -1)/3-1
Sn = 2. (729 -1)/2
Sn = 2. 728/2
Sn = 728
d) (1/5, 2/15, 4/45,...)
q = 2/15 : 1/5 = 2/15 . 5/1 = 2/3
Sn = 1/5 . [(2/3)⁶ - 1]/2/3 - 1
Sn = 1/5 . [64/46656 - 1] /-1/3
Sn = 1/5 . (64/729 -1)/-1/3
Sn = 1/3 . (64-729/729)/-1/3
Sn = 1/3 . - 665/729
Sn = -665/2187 . (-3)
Sn = - 1995/2187 ⇒ simplifica por 3
Sn = -665/729
e) (4,2/3,1/9,...)
q = 2/3 : 4 = 2/3 . 1/4 = 2/12 = 1/6
Sn = 4. [(1/6)⁶ - 1]/1/6 - 1
Sn = 4. [1/46656 -1]/-5/6
Sn = 4. [-46655/46656]/-5/6
Sn = 1 119 720/5
Sn = 223 944
f) (1/3, -1, 3,...)
q= -1 : 1/3 = -3
Sn = 1/3 . [(-3)⁶ - 1]/-3-1
Sn = 1/3 . [729 -1]/-4
Sn = 1/3 . 728/-4
Sn = 728/3 :-4
Sn = 728/3 . - 1/4
Sn = -728/12 (simplifica por 4)
Sn = - 182/3
Resposta. A:0 B:63 C:128 D:6665/729 E:223/944 F:182/3. :a) (√3,√3,√3 ,...)
q = 1
Sn = a1 . (qⁿ - 1)/q - 1
Sn = √3 (1⁶ -1)/1-1
Sn = √3 . 0/0
Sn = 0
b) (32,16,8, ...)
q= 16/32 ∴q = 1/2
Sn = 32 . [(1/2)⁶ - 1]/1/2 - 1
Sn = 32. [1/64 - 1]/-1/2
Sn = 32. [- 63/64]/-1/2
Sn = - 2016/64/-1/2
Sn = -2016/64 . -2/1
Sn = 4032/64
Sn = 63
c) (2,6,18,...)
q= 6:2 = 3
Sn = 2. (3⁶ -1)/3-1
Sn = 2. (729 -1)/2
Sn = 2. 728/2
Sn = 728
d) (1/5, 2/15, 4/45,...)
q = 2/15 : 1/5 = 2/15 . 5/1 = 2/3
Sn = 1/5 . [(2/3)⁶ - 1]/2/3 - 1
Sn = 1/5 . [64/46656 - 1] /-1/3
Sn = 1/5 . (64/729 -1)/-1/3
Sn = 1/3 . (64-729/729)/-1/3
Sn = 1/3 . - 665/729
Sn = -665/2187 . (-3)
Sn = - 1995/2187 ⇒ simplifica por 3
Sn = -665/729
e) (4,2/3,1/9,...)
q = 2/3 : 4 = 2/3 . 1/4 = 2/12 = 1/6
Sn = 4. [(1/6)⁶ - 1]/1/6 - 1
Sn = 4. [1/46656 -1]/-5/6
Sn = 4. [-46655/46656]/-5/6
Sn = 1 119 720/5
Sn = 223 944
f) (1/3, -1, 3,...)
q= -1 : 1/3 = -3
Sn = 1/3 . [(-3)⁶ - 1]/-3-1
Sn = 1/3 . [729 -1]/-4
Sn = 1/3 . 728/-4
Sn = 728/3 :-4
Sn = 728/3 . - 1/4
Sn = -728/12 (simplifica por 4)
Sn = - 182/3