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Jika hasil dari i = 5 ∑ 100 ​ ( ​ ​ 1 + i − 2 ​ ​ 7 ​ ​ ​ ​ − ​ ​ ​ 1 + i − 1 ​ ​ 9 ​ ​ ​ ​ + ​ ​ ​ 1 + i ​ ​ 2 ​ ​ ) adalah ​ ​ 1 + a ​ ​ 7 ​ ​ ​ ​ − ​ ​ ​ 1 + a b ​ ​ 7 ​ ​ ​ ​ − ​ ​ ​ a ​ 2 ​ ​ ​ ​ + ​ ​ ​ b ​ 2 ​ ​ , maka hasil dari i = a ∑ b ​ ( a − b ) adalah ....

Jika hasil dari  adalah , maka hasil dari  adalah ....

  1. negative 72

  2. negative 64

  3. negative 8

  4. 64

  5. 72

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M. Mariyam

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

jawaban yang tepat adalah A.

Pembahasan

Diberikan bentuk sigma sebagai berikut. Terlebih dahulu, jabarkan bentuk notasi sigmanya untuk melihat polanya. Untuk , diperoleh hasil sebagai berikut. Untuk , diperoleh hasil sebagai berikut. Untuk , diperoleh hasil sebagai berikut. Untuk , diperoleh hasil sebagai berikut. Untuk , diperoleh hasil sebagai berikut. Untuk , diperoleh hasil sebagai berikut. Perhatikan bahwa suku-suku yang membentuk diagonal 3 suku akan saling menghabiskan. Oleh karena itu, tersisa perhitungan sebagai berikut. Karena hasil dari adalah , maka didapatkan hubungan sebagai berikut. Dari persamaan di atas, maka nilai dan yang memenuhi adalahsebagai berikut. Dengan demikian, hasil dari dapat ditentukan sebagai berikut. Jadi, jawaban yang tepat adalah A.

Diberikan bentuk sigma sebagai berikut.

begin mathsize 12px style sum from i equals 5 to 100 of open parentheses table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of i minus 2 end root end style end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 9 over denominator square root of 1 plus square root of i minus 1 end root end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of 1 plus square root of i end root end fraction end cell end table close parentheses end style

Terlebih dahulu, jabarkan bentuk notasi sigmanya untuk melihat polanya.

Untuk i equals 5, diperoleh hasil sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus square root of 5 minus 2 end root end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 5 minus 1 end root end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 5 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of 1 plus square root of 3 end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 4 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 5 end root end fraction end cell end table end style

Untuk i equals 6, diperoleh hasil sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus square root of 6 minus 2 end root end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 6 minus 1 end root end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 6 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of 1 plus square root of 4 end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 5 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 6 end root end fraction end cell end table end style

Untuk i equals 7, diperoleh hasil sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus square root of 7 minus 2 end root end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 7 minus 1 end root end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 7 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of 1 plus square root of 5 end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 6 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 7 end root end fraction end cell end table end style

vertical ellipsis

Untuk i equals 98, diperoleh hasil sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus square root of 98 minus 2 end root end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 98 minus 1 end root end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 98 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of 1 plus square root of 96 end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 97 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 98 end root end fraction end cell end table end style

Untuk i equals 99, diperoleh hasil sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus square root of 99 minus 2 end root end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 99 minus 1 end root end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 99 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of 1 plus square root of 97 end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 98 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 99 end root end fraction end cell end table end style

Untuk i equals 100, diperoleh hasil sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus square root of 100 minus 2 end root end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 100 minus 1 end root end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 100 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of 1 plus square root of 98 end root end fraction minus fraction numerator 9 over denominator square root of 1 plus square root of 99 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 100 end root end fraction end cell end table end style

Perhatikan bahwa suku-suku yang membentuk diagonal 3 suku akan saling menghabiskan. Oleh karena itu, tersisa perhitungan sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of 3 end style end root end fraction minus fraction numerator 9 over denominator square root of begin display style 1 plus square root of 4 end style end root end fraction plus fraction numerator 7 over denominator square root of begin display style 1 plus square root of 4 end style end root end fraction plus end cell row blank blank cell fraction numerator 2 over denominator square root of begin display style 1 plus square root of 99 end style end root end fraction minus fraction numerator 9 over denominator square root of begin display style 1 plus square root of 99 end style end root end fraction plus fraction numerator 2 over denominator square root of begin display style 1 plus square root of 100 end style end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of 3 end style end root end fraction minus fraction numerator 2 over denominator square root of 1 plus square root of 4 end root end fraction minus fraction numerator 7 over denominator square root of 1 plus square root of 99 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus square root of 100 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of 3 end style end root end fraction minus fraction numerator 2 over denominator square root of 1 plus 2 end root end fraction minus fraction numerator 7 over denominator square root of 1 plus 3 square root of 11 end root end fraction plus fraction numerator 2 over denominator square root of 1 plus 10 end root end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of 3 end style end root end fraction minus fraction numerator 2 over denominator square root of 3 end fraction minus fraction numerator 7 over denominator square root of 1 plus 3 square root of 11 end root end fraction plus fraction numerator 2 over denominator square root of 11 end fraction end cell row blank equals cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of 3 end style end root end fraction minus fraction numerator 7 over denominator square root of 1 plus 3 square root of 11 end root end fraction minus fraction numerator 2 over denominator square root of 3 end fraction plus fraction numerator 2 over denominator square root of 11 end fraction end cell end table end style

Karena hasil dari begin mathsize 12px style sum from i equals 5 to 100 of open parentheses table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of i minus 2 end root end style end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 9 over denominator square root of 1 plus square root of i minus 1 end root end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of 1 plus square root of i end root end fraction end cell end table close parentheses end style adalah begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of a end style end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus a square root of b end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of a end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of b end fraction end cell end table end style, maka didapatkan hubungan sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of a end style end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus a square root of b end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of a end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of b end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of begin display style 1 plus square root of 3 end style end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 over denominator square root of 1 plus 3 square root of 11 end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of 3 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator square root of 11 end fraction end cell end table end style

Dari persamaan di atas, maka nilai a dan b yang memenuhi adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row a equals 3 row b equals 11 end table

Dengan demikian, hasil dari sum from i equals a to b of open parentheses a minus b close parentheses dapat ditentukan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sum from i equals a to b of open parentheses a minus b close parentheses end cell equals cell sum from i equals 3 to 11 of open parentheses 3 minus 11 close parentheses end cell row blank equals cell sum from i equals 3 to 11 of open parentheses negative 8 close parentheses end cell row blank equals cell 9 cross times open parentheses negative 8 close parentheses end cell row blank equals cell negative 72 end cell end table

Jadi, jawaban yang tepat adalah A.

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