Jika rasio suaru deret geometri tak hingga yang konvergen dan  jumlah deret geometri tak hingga 3+r1​+(3+r)21​+(3+r)31​+..., maka memenuhi nilai ...

Pertanyaan

Jika rasio suaru deret geometri tak hingga yang konvergen dan s jumlah deret geometri tak hingga fraction numerator 1 over denominator 3 plus r end fraction plus 1 over left parenthesis 3 plus r right parenthesis squared plus 1 over left parenthesis 3 plus r right parenthesis cubed plus..., maka s memenuhi nilai ...

  1. 1 fourth less than s less than 1 half 

  2. 3 over 8 less than s less than 3 over 4 

  3. 1 third less than s less than 1 

  4. 3 over 4 less than s less than 4 over 3 

  5. 1 fifth less than s less than 4 over 5 

I. Sutiawan

Master Teacher

Mahasiswa/Alumni Universitas Pasundan

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Ingat!

Rumus rasio barisan geometri:

  • table attributes columnalign right center left columnspacing 0px end attributes row r equals cell U subscript n plus 1 end subscript over U subscript n end cell end table  

Rumus deret geometri tak hingga:

  • table attributes columnalign right center left columnspacing 0px end attributes row cell S subscript infinity end cell equals cell fraction numerator begin display style a end style over denominator 1 minus begin display style r end style end fraction end cell end table 

Syarat deret geometri tak hingga konvergen:

  • negative 1 less than r less than 1 

Diketahui deret geometri tak hingga fraction numerator 1 over denominator 3 plus r end fraction plus 1 over left parenthesis 3 plus r right parenthesis squared plus 1 over left parenthesis 3 plus r right parenthesis cubed plus....

Maka:

table attributes columnalign right center left columnspacing 0px end attributes row r equals cell U subscript n plus 1 end subscript over U subscript n end cell row blank equals cell U subscript 2 over U subscript 1 end cell row blank equals cell fraction numerator begin display style 1 over left parenthesis 3 plus r right parenthesis squared end style over denominator begin display style fraction numerator 1 over denominator open parentheses 3 plus r close parentheses end fraction end style end fraction end cell row blank equals cell fraction numerator 1 over denominator 3 plus r end fraction end cell end table 

Sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell S subscript infinity end cell equals cell fraction numerator begin display style fraction numerator 1 over denominator 3 plus r end fraction end style over denominator 1 minus begin display style fraction numerator 1 over denominator 3 plus r end fraction end style end fraction end cell row blank equals cell fraction numerator 1 over denominator 3 plus r end fraction cross times fraction numerator 3 plus r over denominator 2 plus r end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 plus r end fraction end cell row s equals cell fraction numerator 1 over denominator 2 plus r end fraction end cell end table  

Maka:

 table attributes columnalign right center left columnspacing 0px end attributes row s equals cell fraction numerator 1 over denominator 2 plus r end fraction end cell row cell s left parenthesis 2 plus r right parenthesis end cell equals 1 row cell 2 s plus r s end cell equals 1 row cell r s end cell equals cell 1 minus 2 s end cell row r equals cell fraction numerator 1 minus 2 s over denominator s end fraction end cell end table  

Syarat konvergen negative 1 less than r less than 1, sehingga:

negative 1 less than fraction numerator 1 minus 2 s over denominator s end fraction less than 1 minus s less than 1 minus 2 s less than s minus s less than 1 minus 2 s space dan space 1 minus 2 s less than s minus s plus 2 s less than 1 space dan space minus 2 s minus s less than negative 1 s less than 1 space dan space minus 3 s less than negative 1 s less than 1 space dan space s greater than 1 third 1 third less than s less than 1  

Oleh karena itu, jawaban yang tepat adalah C.

26

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Suku ketiga dan suku keenam dari suatu barisan geometri berturut-turut adalah 6 dan 43​. Jumlah tak hingga deret geometri tersebut adalah ...

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