Question

Wyznacz a1 i q ciągu geometrycznego gdy a2 = 2/9 i a5 = 6.Oblicz S5

Answer 1

Dane:

a₂ = 2/9
a₅ = 6
a₁ = ?
q = ?
S₅ = ?

a₁q = 2/9
a₁q⁴ = 6

a₁q *q³ = 6

2/9*q³ = 6 /*9

2q³ = 54 /:2

q³ = 27

q = - 3 lub q = 3

a₁ = 2/9: (-3) lub a₁ = 2/9: 3

q = - 3 lub q = 3

a₁ =- 2/27 lub a₁ = 2/27

S₅ = a₁ *(1 - q⁵)/(1 - q)

S₅ = (- 2/27)*[ 1 - (-3)⁵]/( 1 + 3) lub S₅ = (2/27)*(1 - 3⁵)/( 1 - 3)

S₅ = (- 2/27)*(1 + 243)/( 1 + 3) lub S₅ = (2/27)*(1 - 243)/( 1 - 3)

S₅ = (- 2/27)*244)/4 lub S₅ = (2/27)*(242)/2

S₅ = (- 2/27)*61 lub S₅ = (2/27)*121

S₅ = -122/27 lub S₅ = 242/27

Answer 2

A₂=²/₉
a₅=6

a₅=a₂×q³
6=²/₉×q³
q³=6:²/₉
q³=27
q=3

a₂=a₁×q
²/₉=a₁×3
a₁=²/₉:3
a₁=²/₂₇

S₅=a₁(1-q⁵):(1-q)=²/₂₇×(1-3⁵):(1-3)=²⁴²/₂₇=8²⁶/₂₇