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Pada suatu deret geometri, diketahui , , dan . Tentukan suku ke-9 deret tersebut.

Pertanyaan

Pada suatu deret geometri, diketahui begin mathsize 14px style a equals 18 end stylebegin mathsize 14px style U subscript n equals 1 half end style, dan begin mathsize 14px style S subscript n equals 15 1 half end style. Tentukan suku ke-9 deret tersebut.

Pembahasan Soal:

Suku ke-n pada barisan geometri, yaitu

begin mathsize 14px style text U end text subscript n equals a times r to the power of n minus 1 end exponent end style

Jumlah n suku pertama deret geometri, yaitu

begin mathsize 14px style S subscript n equals fraction numerator a open parentheses 1 minus text r end text to the power of n close parentheses over denominator 1 minus text r end text end fraction end style untuk begin mathsize 14px style negative 1 less than text r end text less than 1 end style

Diketahui begin mathsize 14px style a equals 18 end stylebegin mathsize 14px style U subscript n equals 1 half end style, dan begin mathsize 14px style S subscript n equals 15 1 half end style sehingga diperoleh

Persamaan (1):

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell text U end text subscript n end cell equals cell a r to the power of n minus 1 end exponent end cell row cell 1 half end cell equals cell 18 times r to the power of n over r end cell row r equals cell 36 r to the power of n end cell end table end style

Persamaan (2):

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell S subscript n end cell equals cell fraction numerator a left parenthesis 1 minus text r end text to the power of n right parenthesis over denominator 1 minus text r end text end fraction end cell row cell 31 over 2 end cell equals cell fraction numerator 18 open parentheses 1 minus r to the power of n close parentheses over denominator 1 minus r end fraction end cell row cell 31 over 36 end cell equals cell fraction numerator left parenthesis 1 minus r to the power of n right parenthesis over denominator open parentheses 1 minus r close parentheses end fraction end cell row cell negative 36 r to the power of n plus 36 end cell equals cell negative 31 r plus 31 end cell row cell negative 36 r to the power of n plus 31 r end cell equals cell negative 5 end cell row cell 36 r to the power of n minus 31 r end cell equals 5 end table end style

Substitusi persamaan (1) ke persamaan (2) diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 36 r to the power of n minus 31 r end cell equals 5 row cell r minus 31 r end cell equals 5 row cell negative 30 r end cell equals 5 row r equals cell negative 5 over 30 end cell row r equals cell negative 1 over 6 end cell end table end style

Dapat ditentukan suku ke-9, yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell U subscript 9 end cell equals cell a times r to the power of 8 end cell row blank equals cell 18 times open parentheses negative 1 over 6 close parentheses to the power of 8 end cell row blank equals cell fraction numerator 18 over denominator 1.679.616 end fraction end cell row blank equals cell fraction numerator 1 over denominator 93.312 end fraction end cell end table end style

Dengan demikian, suku ke-9 deret tersebut adalah begin mathsize 14px style fraction numerator 1 over denominator 93.312 end fraction end style 

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 13 Maret 2021

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