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Suatu barisan geometri tak hingga semua sukunya positif. Jika nilai U3​ + U4​U1​+U2​​ = 9,  tentukan nilai U2​ + U3​U1 ​+ U2 ​+ U3​ + U4 ​+ ...​ .

Pertanyaan

Suatu barisan geometri tak hingga semua sukunya positif. Jika nilai begin mathsize 14px style fraction numerator U subscript 1 plus U subscript 2 over denominator U subscript 3 space plus space U subscript 4 end fraction space equals space 9 comma end style  tentukan nilai begin mathsize 14px style fraction numerator U subscript 1 space end subscript plus space U subscript 2 space end subscript plus space U subscript 3 space plus space U subscript 4 space end subscript plus space... over denominator U subscript 2 space plus space U subscript 3 end fraction end style .

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

nilai dari begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator U subscript 1 plus U subscript 2 plus U subscript 3 plus U subscript 4 plus.... over denominator U subscript 2 plus U subscript 3 end fraction end cell end table equals 27 over 8 end style   

Pembahasan

Ingat rumus untuk menentukan suku ke begin mathsize 14px style n end style barisan geometri begin mathsize 14px style U subscript n end style dan jumlah deret geometri tak hingga begin mathsize 14px style S subscript infinity end style adalah 

begin mathsize 14px style U subscript n equals a r to the power of n minus 1 end exponent end style dan begin mathsize 14px style S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction end style 

Diketahui begin mathsize 14px style fraction numerator U subscript 1 plus U subscript 2 over denominator U subscript 3 plus U subscript 4 end fraction equals 9 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator U subscript 1 plus U subscript 2 over denominator U subscript 3 plus U subscript 4 end fraction end cell equals 9 row cell fraction numerator a plus a r over denominator a r squared plus a r cubed end fraction end cell equals 9 row cell fraction numerator a open parentheses 1 plus r close parentheses over denominator a r squared open parentheses 1 plus r squared close parentheses end fraction end cell equals 9 row cell 1 over r squared end cell equals 9 row cell r squared end cell equals cell 1 over 9 end cell row r equals cell square root of 1 over 9 end root end cell row r equals cell plus-or-minus 1 third end cell end table end style 

Karena barisan geometri tak hingga, semua sukunya positif, maka diambil begin mathsize 14px style r equals 1 third end style 

dengan mensubtitusikan nilai begin mathsize 14px style r equals 1 third end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator U subscript 1 plus U subscript 2 plus U subscript 3 plus U subscript 4 plus.... over denominator U subscript 2 plus U subscript 3 end fraction end cell equals cell fraction numerator S subscript infinity over denominator a r plus a r squared end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator a over denominator 1 minus r end fraction end style over denominator a r open parentheses 1 plus r close parentheses end fraction end cell row blank equals cell fraction numerator a over denominator a r open parentheses 1 plus r close parentheses open parentheses 1 minus r close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator r open parentheses 1 plus r close parentheses open parentheses 1 minus r close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style 1 third end style open parentheses 1 plus begin display style 1 third end style close parentheses open parentheses 1 minus begin display style 1 third end style close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style 1 third end style open parentheses begin display style 4 over 3 end style close parentheses open parentheses begin display style 2 over 3 end style close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style 8 over 27 end style end fraction end cell row blank equals cell 27 over 8 end cell row blank blank blank end table end style 

Jadi nilai dari begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator U subscript 1 plus U subscript 2 plus U subscript 3 plus U subscript 4 plus.... over denominator U subscript 2 plus U subscript 3 end fraction end cell end table equals 27 over 8 end style   

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Intan Retno Wulandari

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Pertanyaan serupa

Jumlah tak hingga dari suatu deret geometri adalah 8 dan rasionya 41​. Besar suku pertama dari deret geometri tersebut adalah ....

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