Question

Sa se completeze căsuțele goale din următorul tabel, știind că (bn)n>1 este progresie geometrică de ratie q

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Answer 1

Răspuns:

Explicație pas cu pas:

progresie geometrica

a) b₁ = 1 ; q = 3 ; n = 4 ; b₄ = ? ; S₄ = ?

b₄ = b₁·q³ = 1·3³ = > b₄ = 27

S₄ = b₁+b₂+b₃+b₄

S₄ = b₁+b₁q+b₁q²+b₁q³= 1+3+9+27 = > S₄ = 40

b) b₁ = 2 ; q = -1 ; n = 5 ; b₅ = ? S₅ = ?

b₅ = b₁·q₄ = 2·(-1)⁴ = > b₅ = 2

b₂ = b₁q = -2 ; b₃ = b₂q = 2 ; b₄ = b₃q = -2

S₅ = 2-2+2-2+2 = > S₅ = 2

c) b₁ = 4 ; q = 1/2 ; bₙ = 1/8 ; n = ? Sₙ = ?

b₂ = b₁q = 4·1/2 = 2 ; b₃ = b₂q = 2·1/2 = 1

b₄ = b₃q = 1/2 ; b₅ = b₄q = 1/2 · 1/2 = 1/4

b₆ = b₅q = 1/4 · 1/2 = 1/8 => n = 6

S₆ = 4+2+1+⁴⁾1/2+²⁾1/4+1/8 = ⁸⁾7 + 7/8 => S₆ = 63/8

d) b₁ = 1 ; n = 8 ; b₈ = -128 ; q = ? S₈ = ?

b₈ = b₁q⁷ => q⁷ = b₈/b₁ = -128 = -2⁷ => q = -2

b₁ = 1 ; b₂ = -2 ; b₃ = 4 ; b₄ = -8 ; b₅ = 16 ;

b₆ = -32 ; b₇ = 64 ; b₈ = -128

S₈ = 1-2+4-8+16-32+64-128 = -1-4-16-64 = > S₈ = -85

e) q = 1/3 ; bₙ = 1/81 ; n = 4 => b₄ = 1/81 ; b₁ = ? ; S₄ = ?

b₃ = b₄/q = 1/81 : 1/3 = 3/81 = 1/27

b₂ = b₃/q = 1/27 : 1/3 = 3/27 = 1/9

b₁ = b₂/q = 1/9 : 1/3 = 3/9 = 1/3

b₁ = 1/3

S₄ = ²⁷⁾1/3 + ⁹⁾1/9 + ³⁾1/27 + 1/81 = (27+9+3+1)/81 => S₄ = 40/81

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