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Hiltunglah jumlah 10 suku pertama dari deret bilangan yang diketahui rumus suku ke-n berikut. 

b.  begin mathsize 14px style U subscript n space equals space 4 space cross times space 3 to the power of 4 minus n end exponent space end style 

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

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Jawaban

 Jumlah undefined suku pertama deret tersebut adalah begin mathsize 14px style 2 over 729 open parentheses 3 to the power of 10 minus 1 close parentheses equals 161 comma 99 almost equal to 162 end style  

Pembahasan

Diketahui begin mathsize 14px style U subscript n equals 4 cross times 3 to the power of 4 minus n end exponent end style 

maka 

begin mathsize 14px style U subscript 1 equals 4 cross times 3 to the power of 4 minus 1 end exponent equals 4 cross times 3 cubed equals 4 cross times 27 equals 108 U subscript 2 equals 4 cross times 3 to the power of 4 minus 2 end exponent equals 4 cross times 3 squared equals 4 cross times 9 equals 36 U subscript 3 equals 4 cross times 3 to the power of 4 minus 3 end exponent equals 4 cross times 3 to the power of 1 equals 4 cross times 3 equals 12 end style 

Barisan di atas merupakan barisan geometri dengan 

begin mathsize 14px style r equals U subscript 2 over U subscript 1 equals 36 over 108 equals 1 third end style 

Jumlah begin mathsize 14px style 10 end style suku pertama deret tersebut adalah 

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 open parentheses 1 minus r to the power of n close parentheses over denominator 1 minus r end fraction end cell row cell S subscript 10 end cell equals cell fraction numerator 108 open square brackets 1 minus open parentheses begin display style 1 third end style close parentheses to the power of 10 close square brackets over denominator 1 minus begin display style 1 third end style end fraction end cell row blank equals cell fraction numerator 108 open parentheses 1 minus begin display style 1 over 3 to the power of 10 end style close parentheses over denominator begin display style 2 over 3 end style end fraction end cell row blank equals cell 108 open parentheses 3 over 2 close parentheses open parentheses fraction numerator 3 to the power of 10 minus 1 over denominator 3 to the power of 10 end fraction close parentheses end cell row blank equals cell fraction numerator 4 open parentheses 27 close parentheses open parentheses 3 close parentheses open parentheses 3 to the power of 10 minus 1 close parentheses over denominator open parentheses 2 close parentheses open parentheses 3 to the power of 10 close parentheses end fraction end cell row blank equals cell fraction numerator 2 open parentheses 3 to the power of 4 close parentheses open parentheses 3 to the power of 10 minus 1 close parentheses over denominator 3 to the power of 10 end fraction end cell row blank equals cell 2 over 3 to the power of 6 open parentheses 3 to the power of 10 minus 1 close parentheses end cell row blank equals cell 2 over 729 open parentheses 3 to the power of 10 minus 1 close parentheses end cell row blank equals cell 161 comma 99 almost equal to 162 end cell end table end style 

Jadi Jumlah undefined suku pertama deret tersebut adalah begin mathsize 14px style 2 over 729 open parentheses 3 to the power of 10 minus 1 close parentheses equals 161 comma 99 almost equal to 162 end style  

36

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