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Diketahui suatu deret geometri dengan U2​=6 dan U4​=54. Tentukan jumlah 5 suku pertama deret tersebut.

Pertanyaan

Diketahui suatu deret geometri dengan begin mathsize 14px style straight U subscript 2 equals 6 end style dan begin mathsize 14px style straight U subscript 4 equals 54 end style. Tentukan jumlah 5 suku pertama deret tersebut.

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

 jumlah 5 suku pertama deret tersebut adalah 242 atau negative 122.

Pembahasan

Karena diketahui deret geometri, maka rumus suku ke-n adalah begin mathsize 14px style straight U subscript straight n equals ar to the power of straight n minus 1 end exponent end style . Tentukan nlai begin mathsize 14px style straight r end style sebagai berikut;

table attributes columnalign right center left columnspacing 0px end attributes row cell straight U subscript 2 end cell equals 6 row cell ar to the power of 2 minus 1 end exponent end cell equals 6 row ar equals 6 row blank blank blank row cell straight U subscript 4 end cell equals 54 row cell ar to the power of 4 minus 1 end exponent end cell equals 54 row cell ar cubed end cell equals 54 row cell open parentheses ar close parentheses straight r squared end cell equals 54 row cell 6 straight r squared end cell equals 54 row cell straight r squared end cell equals 9 row straight r equals cell plus-or-minus 3 end cell row r equals cell 3 space a t a u space r equals negative 3 end cell end table 

maka nilai begin mathsize 14px style straight a end style sebagi berikut;

table attributes columnalign right center left columnspacing 0px end attributes row ar equals 6 row cell straight a cross times 3 end cell equals 6 row straight a equals 2 end table  atau  table attributes columnalign right center left columnspacing 0px end attributes row ar equals 6 row cell straight a cross times left parenthesis negative 3 right parenthesis end cell equals 6 row straight a equals cell negative 2 end cell end table

Jumlah 5 suku pertama deret geometri adalah sebagai berikut;

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight S subscript straight n end cell equals cell fraction numerator straight a open parentheses straight r to the power of straight n minus 1 close parentheses over denominator straight r minus 1 end fraction end cell row cell straight S subscript 5 end cell equals cell fraction numerator 2 open parentheses 3 to the power of 5 minus 1 close parentheses over denominator 3 minus 1 end fraction end cell row cell straight S subscript 5 end cell equals cell fraction numerator 2 open parentheses 243 minus 1 close parentheses over denominator 2 end fraction end cell row cell straight S subscript 5 end cell equals cell fraction numerator 2 open parentheses 242 close parentheses over denominator 2 end fraction end cell row cell straight S subscript 5 end cell equals 242 end table end style  atau table attributes columnalign right center left columnspacing 0px end attributes row cell straight S subscript straight n end cell equals cell fraction numerator straight a open parentheses straight r to the power of straight n minus 1 close parentheses over denominator straight r minus 1 end fraction end cell row cell straight S subscript 5 end cell equals cell fraction numerator negative 2 open parentheses left parenthesis negative 3 right parenthesis to the power of 5 minus 1 close parentheses over denominator left parenthesis negative 3 right parenthesis minus 1 end fraction end cell row cell straight S subscript 5 end cell equals cell fraction numerator negative 2 open parentheses left parenthesis negative 243 right parenthesis minus 1 close parentheses over denominator negative 4 end fraction end cell row cell straight S subscript 5 end cell equals cell fraction numerator negative 2 open parentheses negative 244 close parentheses over denominator negative 4 end fraction end cell row cell straight S subscript 5 end cell equals cell negative 122 end cell end table

Sehingga, jumlah 5 suku pertama deret tersebut adalah 242 atau negative 122.

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