Roboguru

begin mathsize 14px style 8 plus 7 plus 9 plus 3 plus horizontal ellipsis plus 1 over 27 plus 1 over 81 equals horizontal ellipsis end style 

  1. begin mathsize 14px style horizontal ellipsis end style 

  2. begin mathsize 14px style horizontal ellipsis end style 

Jawaban:

Suku kebegin mathsize 14px style negative 3 end style sampai dengan suku terakhir dari deret di atas merupakan deret geometri.

Rasio dari deret tersebut adalah begin mathsize 14px style r equals 3 over 9 equals 1 third end style. Untuk suku pertama begin mathsize 14px style a equals 9 end style dan suku terakhir begin mathsize 14px style U subscript n equals 1 over 81 end style, jumlah suku dari deret tersebut adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell U subscript n end cell equals cell a r to the power of n minus 1 end exponent end cell row cell 1 over 81 end cell equals cell 9 times open parentheses 1 third close parentheses to the power of n minus 1 end exponent end cell row cell 3 to the power of negative 4 end exponent end cell equals cell 3 squared times open parentheses 3 to the power of negative 1 end exponent close parentheses to the power of n minus 1 end exponent end cell row cell 3 to the power of negative 4 end exponent end cell equals cell 3 squared times 3 to the power of negative n plus 1 end exponent end cell row cell 3 to the power of negative 4 end exponent end cell equals cell 3 to the power of 2 minus n plus 1 end exponent end cell row cell 3 to the power of negative 4 end exponent end cell equals cell 3 to the power of 3 minus n end exponent end cell row cell negative 4 end cell equals cell 3 minus n end cell row n equals cell 3 plus 4 end cell row n equals 7 end table end style 

Jumlahan dari  begin mathsize 14px style 7 end style suku pada deret untuk begin mathsize 14px style r less than 1 end style 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 7 end cell equals cell fraction numerator 9 times open parentheses 1 minus open parentheses begin display style 1 third end style close parentheses to the power of 7 close parentheses over denominator 1 minus begin display style 1 third end style end fraction end cell row blank equals cell fraction numerator 9 times open parentheses 1 minus begin display style 1 over 2187 end style close parentheses over denominator begin display style 2 over 3 end style end fraction end cell row blank equals cell 9 times 3 over 2 times 2186 over 2187 end cell row blank equals cell 1093 over 81 end cell end table end style 

Dengan demikian, hasil dari jumlahan pada deret di atas yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 8 plus 7 plus 9 plus 3 plus horizontal ellipsis plus 1 over 27 plus 1 over 81 end cell row blank equals cell 8 plus 7 plus S subscript 7 end cell row blank equals cell 15 plus 1093 over 81 end cell row blank equals cell 2308 over 81 end cell end table end style 

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