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Pertanyaan

Diketahui S adalah jumlah deret geometri tak hingga begin mathsize 14px style fraction numerator 1 over denominator 3 plus r end fraction plus 1 over open parentheses 3 plus r close parentheses squared plus 1 over open parentheses 3 plus r close parentheses cubed plus... end style dan begin mathsize 14px style negative 1 less than r less than 1 end style  rasionya. Jika deret tersebut konvergen maka . . . .

  1. size 14px 1 over size 14px 4 size 14px less than size 14px S size 14px less than size 14px 1 over size 14px 2

  2. size 14px 3 over size 14px 8 size 14px less than size 14px S size 14px less than size 14px 3 over size 14px 4 

  3. size 14px 1 over size 14px 3 size 14px less than size 14px S size 14px less than size 14px 1

  4. size 14px 3 over size 14px 4 size 14px less than size 14px S size 14px less than size 14px 4 over size 14px 3

  5. size 14px 1 over size 14px 5 size 14px less than size 14px S size 14px less than size 14px 4 over size 14px 5

N. Indah

Master Teacher

Mahasiswa/Alumni Universitas Diponegoro

Jawaban terverifikasi

Jawaban

jawabanyang tepat adalah C.

Pembahasan

begin mathsize 14px style text S end text subscript infinity equals fraction numerator a over denominator 1 minus text r end text end fraction space text dengan end text space minus 1 less than text r end text less than 1 end style

Misal:

Jika kita ambil  begin mathsize 14px style text r=1 end text end style, maka deret geometri tak hingga tersebut menjadi

begin mathsize 14px style fraction numerator 1 over denominator 3 plus 1 end fraction plus 1 over open parentheses 3 plus 1 close parentheses squared plus 1 over open parentheses 3 plus 1 close parentheses cubed plus midline horizontal ellipsis equals 1 fourth plus 1 over 16 plus 1 over 64 plus horizontal ellipsis end style

Diperoleh begin mathsize 14px style text a= end text 1 fourth space text dan r end text equals 1 fourth end style

begin mathsize 14px style S equals fraction numerator a over denominator 1 minus r end fraction equals fraction numerator begin display style 1 fourth end style over denominator 1 minus begin display style 1 fourth end style end fraction equals fraction numerator begin display style 1 fourth end style over denominator begin display style 3 over 4 end style end fraction equals 3 over 4 end style

Jika kita ambil begin mathsize 14px style text r=-1 end text end style, maka deret geometri tak hingga tersebut menjadi

begin mathsize 14px style fraction numerator 1 over denominator 3 plus open parentheses negative 1 close parentheses end fraction plus 1 over open parentheses 3 plus open parentheses negative 1 close parentheses close parentheses squared plus 1 over open parentheses 3 plus open parentheses negative 1 close parentheses close parentheses cubed plus horizontal ellipsis equals 1 half plus 1 fourth plus 1 over 8 plus horizontal ellipsis end style

Diperoleh begin mathsize 14px style text a= end text 1 half space text dan r end text equals 1 half end style

begin mathsize 14px style S equals fraction numerator a over denominator 1 minus r end fraction equals fraction numerator begin display style 1 half end style over denominator 1 minus begin display style 1 half end style end fraction equals fraction numerator begin display style 1 half end style over denominator begin display style 1 half end style end fraction equals 1 end style

Dari uraian di atas, diperoleh begin mathsize 14px style 1 third less than text S end text less than 1 end style.

Jadi, jawabanyang tepat adalah C.

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Untuk harga-harga x manakah deret geometri tak hingga berikut ini konvergen, jika rasionya sama dengan 3log(x−1).

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