Suatu barisan geometri tak hingga semua sukunya positif. Jika nilai U3 + U4U1+U2 = 9, tentukan nilai U2 + U3U1 + 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 barisan geometri dan jumlah deret geometri tak hingga adalah

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

dengan mensubtitusikan nilai , 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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