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Jika barisan goemetri  mempunyai , maka  adalah ...

Pertanyaan

Jika barisan goemetri begin mathsize 14px style left parenthesis y plus 1 right parenthesis comma space left parenthesis 2 y minus 2 right parenthesis comma space left parenthesis 7 y minus 1 right parenthesis end style mempunyai begin mathsize 14px style r greater than 0 end style, maka begin mathsize 14px style U subscript 4 end style adalah ...

  1. begin mathsize 14px style negative 108 end style 

  2. begin mathsize 14px style 4 over 3 end style 

  3. begin mathsize 14px style negative 3 over 4 end style 

  4. begin mathsize 14px style negative 4 over 3 end style 

  5. begin mathsize 14px style negative 324 end style 

Pembahasan Soal:

Rasio pada barisan geometri merupakan perbandingan dua suku yang berdekatan, hal ini dapat dirumuskan:

begin mathsize 14px style r equals U subscript 2 over U subscript 1 equals U subscript 3 over U subscript 2 equals... equals U subscript n over U subscript n minus 1 end subscript end style 

Diketahui barisan geometri:

begin mathsize 14px style left parenthesis y plus 1 right parenthesis comma space left parenthesis 2 y minus 2 right parenthesis comma space left parenthesis 7 y minus 1 right parenthesis end style

Gunakan rumus rasio untuk mencari nilai begin mathsize 14px style y end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell U subscript 2 over U subscript 1 end cell equals cell U subscript 3 over U subscript 2 end cell row cell fraction numerator 2 y minus 2 over denominator y plus 1 end fraction end cell equals cell fraction numerator 7 y minus 1 over denominator 2 y minus 2 end fraction end cell row cell open parentheses 2 y minus 2 close parentheses squared end cell equals cell open parentheses 7 y minus 1 close parentheses open parentheses y plus 1 close parentheses end cell row cell 4 y squared minus 8 y plus 4 end cell equals cell 7 y squared plus 6 y minus 1 end cell row cell 7 y squared minus 4 y squared plus 6 y plus 8 y minus 1 minus 4 end cell equals 0 row cell 3 y squared plus 14 y minus 5 end cell equals 0 row cell open parentheses 3 y minus 1 close parentheses open parentheses y plus 5 close parentheses end cell equals 0 row y equals cell 1 third space atau space y equals negative 5 end cell end table end style 

Substitusikan nilai y pada ketiga barisan geometri.

Untuk begin mathsize 14px style y equals 1 third end style,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y plus 1 end cell equals cell 1 third plus 1 end cell row blank equals cell 4 over 3 end cell row cell 2 y minus 2 end cell equals cell 2 times 1 third minus 2 end cell row blank equals cell 2 over 3 minus 6 over 3 end cell row blank equals cell negative 4 over 3 end cell row cell 7 y minus 1 end cell equals cell 7 times 1 third minus 1 end cell row blank equals cell 7 over 3 minus 3 over 3 end cell row blank equals cell 4 over 3 end cell end table end style 

maka barisan geometrinya:

begin mathsize 14px style 4 over 3 comma space minus 4 over 3 comma space 4 over 3 comma space... end style 

Untuk size 14px y size 14px equals size 14px minus size 14px 5,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y plus 1 end cell equals cell negative 5 plus 1 end cell row blank equals cell negative 4 end cell row cell 2 y minus 2 end cell equals cell 2 times open parentheses negative 5 close parentheses minus 2 end cell row blank equals cell negative 10 minus 2 end cell row blank equals cell negative 12 end cell row cell 7 y minus 1 end cell equals cell 7 times open parentheses negative 5 close parentheses minus 1 end cell row blank equals cell negative 35 minus 1 end cell row blank equals cell negative 36 end cell end table end style

maka, barisan geometrinya:

begin mathsize 14px style negative 4 comma space minus 12 comma space minus 36 comma space... end style

Carilah rasio barisan geometri tersebut. 

Untuk begin mathsize 14px style y equals 1 third end style, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row r equals cell U subscript 2 over U subscript 1 end cell row r equals cell fraction numerator begin display style bevelled fraction numerator negative 4 over denominator 3 end fraction end style over denominator begin display style bevelled 4 over 3 end style end fraction end cell row r equals cell negative 1 end cell end table end style 

Untuk size 14px y size 14px equals size 14px minus size 14px 5,  maka:

 begin mathsize 14px style r equals U subscript 2 over U subscript 1 r equals fraction numerator negative 12 over denominator negative 4 end fraction r equals 3 end style 

karena begin mathsize 14px style r greater than 0 end style, maka pilih table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px r end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 3 end table.

Sehingga suku ke-4 barisan geometri adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell U subscript n end cell equals cell a r to the power of n minus 1 end exponent end cell row cell U subscript 4 end cell equals cell negative 4 times 3 to the power of 4 minus 1 end exponent end cell row blank equals cell negative 4 times 27 end cell row blank equals cell negative 108 end cell end table end style 

Jadi, jawaban yang tepat adalah A.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 04 Maret 2021

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