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Tentukan jumlah deret geometri berikut .

Tentukan jumlah deret geometri berikut begin mathsize 14px style 48 plus 12 plus 3 plus... plus 3 over 16 end style.space

 

Jawaban:

Pada soal suku pertama adalah begin mathsize 14px style 48 end style, sehingga begin mathsize 14px style a equals 48 end style. Rasio merupakan perbandingan suku ke-dengan suku ke-(n-1), sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row r equals cell U subscript n over U subscript n minus 1 end subscript end cell row blank equals cell U subscript 2 over U subscript 2 minus 1 end subscript end cell row blank equals cell U subscript 2 over U subscript 1 end cell row blank equals cell fraction numerator begin display style 48 end style over denominator begin display style 12 end style end fraction end cell row blank equals cell 1 fourth end cell end table end style

Ingat rumus suku ke-n deret geometri begin mathsize 14px style U subscript n equals a r to the power of n minus 1 end exponent end style, maka banyak suku pada deret tersebut adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell 3 over 16 equals 48 cross times open parentheses 1 fourth close parentheses to the power of n minus 1 end exponent end cell row cell 3 over 768 end cell equals cell open parentheses 1 fourth close parentheses to the power of n minus 1 end exponent end cell row cell 1 over 256 end cell equals cell fraction numerator open parentheses begin display style 1 fourth end style close parentheses to the power of n over denominator open parentheses begin display style 1 fourth end style close parentheses end fraction end cell row cell 1 over 256 end cell equals cell open parentheses 1 fourth close parentheses to the power of n cross times 4 end cell row cell fraction numerator 1 over denominator 1.024 end fraction end cell equals cell 1 to the power of n over 4 to the power of n end cell row cell 4 to the power of n end cell equals cell 1.024 end cell row cell 4 to the power of n end cell equals cell 4 to the power of 5 end cell row blank rightwards arrow cell n equals 5 end cell end table end style

 

Ingat rumus jumlah suku ke-deret geometri begin mathsize 14px style S subscript n equals fraction numerator a left parenthesis 1 minus r to the power of n right parenthesis over denominator left parenthesis 1 minus r right parenthesis end fraction comma space r less than 1 end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell S subscript 5 equals fraction numerator begin display style 48 open parentheses 1 minus open parentheses 1 fourth close parentheses to the power of 5 close parentheses end style over denominator 1 minus open parentheses begin display style 1 fourth end style close parentheses end fraction end cell row blank equals cell fraction numerator begin display style 48 open parentheses 1 minus open parentheses fraction numerator 1 over denominator 1.024 end fraction close parentheses close parentheses end style over denominator begin display style 3 over 4 end style end fraction end cell row blank equals cell fraction numerator begin display style 48 cross times open parentheses fraction numerator 1.023 over denominator 1.024 end fraction close parentheses end style over denominator begin display style 3 over 4 end style end fraction end cell row blank equals cell fraction numerator 49.104 over denominator 1.024 end fraction cross times 4 over 3 end cell row blank equals cell fraction numerator 1.023 over denominator 16 end fraction end cell row blank equals cell 63 15 over 16 end cell end table end style

 

Jadi, jumlah deret geometri tersebut adalah begin mathsize 14px style 63 15 over 16 end style.

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