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Sebuah barisan ditentukan secara rekursif seperti ...

Sebuah barisan ditentukan secara rekursif seperti berikut ini.
begin mathsize 14px style X subscript n equals open curly brackets table row cell 1 semicolon space n equals 1 end cell row cell fraction numerator X subscript n minus 1 end subscript over denominator 1 plus X subscript n minus 1 end subscript end fraction semicolon space n greater than 1 end cell end table close end style 

    Tentukan nilai begin mathsize 14px style X subscript 6 end style sampai begin mathsize 14px style X subscript 10 end style, kemudian tentukan nilai begin mathsize 14px style left parenthesis X subscript 10 minus X subscript 8 right parenthesis end style 

 

Jawaban:

 

 

 barisan ditentukan secara rekursif seperti berikut ini.
begin mathsize 14px style X subscript n equals open curly brackets table row cell 1 semicolon space n equals 1 end cell row cell fraction numerator X subscript n minus 1 end subscript over denominator 1 plus X subscript n minus 1 end subscript end fraction semicolon space n greater than 1 end cell end table close end style 

Pada begin mathsize 14px style X subscript 1 end style sampai begin mathsize 14px style X subscript 5 end style     

  • Untuk begin mathsize 14px style n equals 1 end style, maka begin mathsize 14px style X subscript 1 equals 1 end style

 

  • untuk begin mathsize 14px style n equals 2 end style, maka

  •  begin mathsize 14px style X subscript 2 equals fraction numerator X subscript 1 over denominator 1 plus X subscript 1 end fraction X subscript 2 equals fraction numerator 1 over denominator 1 plus 1 end fraction X subscript 2 equals 1 half end style

 

  • Untuk begin mathsize 14px style n equals 3 end style, maka
  • begin mathsize 14px style X subscript 3 equals fraction numerator X subscript 2 over denominator 1 plus X subscript 2 end fraction X subscript 3 equals fraction numerator begin display style 1 half end style over denominator 1 plus begin display style 1 half end style end fraction X subscript 3 equals fraction numerator begin display style 1 half end style over denominator begin display style 3 over 2 end style end fraction X subscript 3 equals 1 third end style 

 

  • Untuk begin mathsize 14px style n equals 4 end style, maka 
  • begin mathsize 14px style X subscript 4 equals fraction numerator X subscript 3 over denominator 1 plus X subscript 3 end fraction X subscript 4 equals fraction numerator begin display style 1 third end style over denominator 1 plus begin display style 1 third end style end fraction X subscript 4 equals fraction numerator begin display style 1 third end style over denominator begin display style 4 over 3 end style end fraction X subscript 4 equals 1 fourth  end style 

 

  • Untuk undefined, maka 
  • begin mathsize 14px style X subscript 5 equals fraction numerator X subscript 4 over denominator 1 plus X subscript 4 end fraction X subscript 5 equals fraction numerator begin display style 1 fourth end style over denominator 1 plus begin display style 1 fourth end style end fraction X subscript 5 equals 1 fifth end style 

Jika diteruskan nilai begin mathsize 14px style X subscript 6 end style sampai begin mathsize 14px style X subscript 10 end style dan disajikan ke dalam tabel sebagai berikut:

 

Selanjutnya nilai begin mathsize 14px style left parenthesis X subscript 10 minus X subscript 8 right parenthesis end style  adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell X subscript 10 minus X subscript 8 end cell equals cell 1 over 10 minus 1 over 8 end cell row blank equals cell fraction numerator 8 minus 10 over denominator 80 end fraction end cell row blank equals cell negative 2 over 80 end cell row blank equals cell negative 1 over 40 end cell end table end style 

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