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Misalkan  dan . Nilai dari .

Pertanyaan

Misalkan begin mathsize 14px style 1 plus 1 over p plus 1 over p squared plus 1 over p cubed plus space... space equals 2021 end style dan begin mathsize 14px style 1 over q plus 1 over q squared plus 1 over q cubed plus space... space equals 2019 end style. Nilai dari begin mathsize 14px style q over p equals space... end style

  1. begin mathsize 14px style fraction numerator 2018 cross times 2020 over denominator 2021 cross times 2019 end fraction end style 

  2. fraction numerator size 14px 2019 size 14px cross times size 14px 2020 over denominator size 14px 2018 size 14px cross times size 14px 2021 end fraction 

  3. begin mathsize 14px style fraction numerator 2020 squared minus 1 over denominator 2020 squared end fraction end style 

  4. begin mathsize 14px style fraction numerator 2020 squared over denominator 2020 squared minus 1 end fraction end style 

  5. begin mathsize 14px style fraction numerator 2021 squared over denominator 2021 squared minus 1 end fraction end style 

Pembahasan Soal:

Gunakan rumus jumlah deret geometri tak hingga dengan begin mathsize 14px style negative 1 less than r less than 1 end style:

begin mathsize 14px style straight S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction a equals suku space pertama r equals rasio equals straight U subscript n over straight U subscript n minus 1 end subscript end style 

Diketahui:

begin mathsize 14px style 1 plus 1 over p plus 1 over p squared plus 1 over p cubed plus space... space equals 2021 end style

begin mathsize 14px style 1 over q plus 1 over q squared plus 1 over q cubed plus space... space equals 2019 end style

Akan ditentukan nilai dari begin mathsize 14px style q over p end style.

Perhatikan begin mathsize 14px style 1 plus 1 over p plus 1 over p squared plus 1 over p cubed plus space... space equals 2021 end style, untuk menentukan rasio dan suku pertamanya, pada deret tersebut diperoleh suku pertama begin mathsize 14px style a equals 1 end style,  rasio begin mathsize 14px style r equals straight U subscript 2 over straight U subscript 1 equals fraction numerator begin display style 1 over p end style over denominator 1 end fraction equals 1 over p end style serta jumlah deret tak hingga adalah begin mathsize 14px style straight S subscript infinity equals 2021 end style, dengan menggunakan deret geometri tak hingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight S subscript infinity end cell equals cell fraction numerator a over denominator 1 minus r end fraction end cell row 2021 equals cell fraction numerator 1 over denominator 1 minus begin display style 1 over p end style end fraction end cell row cell 1 minus 1 over p end cell equals cell 1 over 2021 end cell row cell 1 minus 1 over 2021 end cell equals cell 1 over p end cell row cell fraction numerator 2021 minus 1 over denominator 2021 end fraction end cell equals cell 1 over p end cell row cell 2020 over 2021 end cell equals cell 1 over p end cell row cell 2021 over 2020 end cell equals cell p over 1 end cell row cell 2021 over 2020 end cell equals p row p equals cell 2021 over 2020 end cell end table end style 

Perhatikan begin mathsize 14px style 1 over q plus 1 over q squared plus 1 over q cubed plus space... space equals 2019 end style, untuk menentukan rasio dan suku pertamanya, pada deret tersebut diperoleh suku pertama begin mathsize 14px style a equals 1 over q end style, rasio begin mathsize 14px style r equals straight U subscript 2 over straight U subscript 1 equals fraction numerator begin display style 1 over q squared end style over denominator begin display style 1 over q end style end fraction equals 1 over q squared cross times q over 1 equals 1 over q end style serta jumlah deret tak hingga adalah begin mathsize 14px style straight S subscript infinity equals 2019 end style, dengan menggunakan deret geometri tak hingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight S subscript infinity end cell equals cell fraction numerator a over denominator 1 minus r end fraction end cell row 2019 equals cell fraction numerator begin display style 1 over q end style over denominator 1 minus begin display style 1 over q end style end fraction end cell row 2019 equals cell fraction numerator begin display style 1 over q end style over denominator begin display style fraction numerator q minus 1 over denominator q end fraction end style end fraction end cell row 2019 equals cell 1 over q cross times fraction numerator q over denominator q minus 1 end fraction end cell row 2019 equals cell fraction numerator 1 over denominator q minus 1 end fraction end cell row cell q minus 1 end cell equals cell 1 over 2019 end cell row q equals cell 1 over 2019 plus 1 end cell row q equals cell fraction numerator 1 plus 2019 over denominator 2019 end fraction end cell row q equals cell 2020 over 2019 end cell end table end style  

Nilai begin mathsize 14px style q over p end style dapat dihitung sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell q over p end cell equals cell fraction numerator begin display style 2020 over 2019 end style over denominator begin display style 2021 over 2020 end style end fraction end cell row blank equals cell 2020 over 2019 cross times 2020 over 2021 end cell row blank equals cell fraction numerator 2020 cross times 2020 over denominator 2019 cross times 2021 end fraction end cell row blank equals cell fraction numerator 2020 squared over denominator open parentheses 2020 minus 1 close parentheses open parentheses 2020 plus 1 close parentheses end fraction end cell row cell q over p end cell equals cell fraction numerator 2020 squared over denominator 2020 squared minus 1 end fraction end cell end table end style 

Jadi, jawaban yang tepat adalah D.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 13 Maret 2021

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