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Diketahui barisan geometri , dengan  dan . Jumlah ...

Diketahui barisan geometri begin mathsize 14px style straight u subscript straight n end style, dengan begin mathsize 14px style straight u subscript 3 plus straight u subscript 4 equals 9 open parentheses straight u subscript 1 plus straight u subscript 2 close parentheses end style dan begin mathsize 14px style straight u subscript 1 times straight u subscript 2 equals 18 straight u subscript 2 end style. Jumlah begin mathsize 14px style 4 end style suku pertama yang mungkin adalah ...

  1. 66space 

  2. 72space 

  3. 78space 

  4. 80space 

  5. 88space 

Jawaban:

Pada barisan dan deret geometri begin mathsize 14px style straight u subscript straight n equals ar to the power of straight n minus 1 end exponent end style, sehingga 

begin mathsize 14px style straight u subscript 2 equals ar straight u subscript 3 equals ar squared straight u subscript 4 equals ar cubed end style 

Maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight u subscript 3 plus straight u subscript 4 end cell equals cell 9 left parenthesis straight u subscript 1 plus straight u subscript 2 right parenthesis end cell row cell ar squared plus ar cubed end cell equals cell 9 left parenthesis straight a plus ar right parenthesis end cell row cell ar squared plus ar cubed end cell equals cell 9 straight a plus 9 ar end cell row cell straight r squared plus straight r cubed end cell equals cell 9 plus 9 straight r end cell row cell straight r cubed plus straight r squared minus 9 straight r minus 9 end cell equals 0 end table end style  

Uji undefined   

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight r cubed plus straight r squared minus 9 straight r minus 9 end cell equals 0 row cell left parenthesis 3 right parenthesis cubed plus left parenthesis 3 right parenthesis squared minus 9 left parenthesis 3 right parenthesis minus 9 end cell equals 0 row cell 27 plus 9 minus 27 minus 9 end cell equals 0 row 0 equals cell 0 space left parenthesis memenuhi right parenthesis end cell end table end style  

Jadi rasio barisan tersebut adalah begin mathsize 14px style straight r equals 3 end style, karena begin mathsize 14px style straight u subscript 1 times straight u subscript 2 equals 18 straight u subscript 2 end style maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight u subscript 1 times straight u subscript 4 end cell equals cell 18 straight u subscript 2 space end cell row cell straight a times ar cubed end cell equals cell 18 times ar end cell row cell straight a squared straight r cubed end cell equals cell 18 ar end cell row cell ar squared end cell equals 18 row cell straight a left parenthesis 3 right parenthesis squared end cell equals 18 row cell 9 straight a end cell equals 18 row straight a equals 2 end table end style  

Maka jumlah 4 suku pertama deret tersebut adalah 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight S subscript straight n end cell equals cell fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction end cell row cell straight S subscript 4 end cell equals cell fraction numerator 2 left parenthesis 3 to the power of 4 minus 1 right parenthesis over denominator 3 minus 1 end fraction end cell row blank equals cell fraction numerator 2 times left parenthesis 81 minus 1 right parenthesis over denominator 2 end fraction end cell row blank equals 80 end table end style   

Jadi jawaban yang tepat adalah D.

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