Question

Să se determine primul termen și rația progresiei aritmetice (an) dacă:

d) {a4+a8=30
{10a1-4a7= -45

e) {a2+a6+a9=45
{a3+a7+a10=54

g) {S2-S4+a2=14
{S3+a3=17
​

Answer 1

Răspuns:

Explicație pas cu pas:

progresii aritmetice

d) {a₄+a₈=30

   {10a₁-4a₇= -45  =>

{a₁+3r +a₁+7r = 30

{10a₁-4(a₁+6r) = -45 <=>

2a₁ + 10r = 30

10a₁-4a₁-24r = -45 =>

2a₁+10r = 30  I·(-3)

6a₁-24r = -45 =>

-6a₁-30r = -90

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-54r = -135  => r = -135/54 => r = -2,5 ;

2a₁+10·(-2,5) = 30 => 2a₁ = 30+25 = 55 => a₁ = 55:2 => a₁ = 27,5

e) {a₂+a₆+a₉= 45   <=> {a₁+r+a₁+5r+a₁+8r = 45

    {a₃+a₇+a₁₀=54        {a₁+2r+a₁+6r+a₁+9r = 54  =>

{3a₁+14r = 45

{3a₁+17r = 54

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3r = 54-45 = 9 => r = 9:3 => r = 3

3a₁+14·3 = 45 <=> 3a₁ = 45-42 = 3 => a₁ = 1

g) {S₂-S₄+a₂=14       {a₁+a₁+r -(a₁+a₁+r+a₁+2r+a₁+3r)+a₁+r = 14

   {S₃+a₃=17   <=>  { a₁+a₁+r+a₁+2r+a₁+2r = 17 =>

{2a₁-4a₁+a₁+r-6r+r = 14     { -a₁-4r = 14   I·4   =>

{4a₁+5r = 17   <=>              {4a₁+5r = 17

{-4a₁-16r = 56

{4a₁+5r = 17

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-11r = 56+17  = 73 => r = -73/11

4a₁+5·(-73/11) = 17  => ¹¹⁾4a₁ = ¹¹⁾17+365/11 = >

44a₁ = 187+365 = 552 => a₁ = 552/44 = > a₁ = 138/11

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