Roboguru

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

Pertanyaan

Jika 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 hasil dari sum from i equals a to b of open parentheses a minus b close parentheses adalah ....

  1. negative 72

  2. negative 64

  3. negative 8

  4. 64

  5. 72

Pembahasan Soal:

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Mariyam

Mahasiswa/Alumni Institut Pertanian Bogor

Terakhir diupdate 30 Agustus 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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Kurang Membantu

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Mau Bertanya

Pertanyaan yang serupa

Hasil dari  adalah ....

Pembahasan Soal:

Perhatikan hubungan berikut!

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator k open parentheses k plus 1 close parentheses end fraction end cell equals cell fraction numerator k plus 1 minus k over denominator k open parentheses k plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator k plus 1 over denominator k open parentheses k plus 1 close parentheses end fraction minus fraction numerator k over denominator k open parentheses k plus 1 close parentheses end fraction end cell row blank equals cell 1 over k minus fraction numerator 1 over denominator k plus 1 end fraction end cell end table end style

Oleh karena itu, didapat perhitungan sebagai berikut.

begin mathsize 12px style sum from k equals 1 to 100 of fraction numerator 1 over denominator k open parentheses k plus 1 close parentheses end fraction equals sum subscript k equals 1 end subscript superscript 100 open parentheses 1 over k minus fraction numerator 1 over denominator k plus 1 end fraction close parentheses equals open parentheses 1 over 1 minus fraction numerator 1 over denominator 1 plus 1 end fraction close parentheses plus open parentheses 1 half minus fraction numerator 1 over denominator 2 plus 1 end fraction close parentheses plus open parentheses 1 third minus fraction numerator 1 over denominator 3 plus 1 end fraction close parentheses plus horizontal ellipsis plus open parentheses 1 over 100 minus fraction numerator 1 over denominator 100 plus 1 end fraction close parentheses equals 1 over 1 minus 1 half plus 1 half minus 1 third plus 1 third minus 1 fourth plus horizontal ellipsis plus 1 over 100 minus 1 over 101 equals 1 over 1 minus 1 over 101 equals 1 minus 1 over 101 end style

Jadi, jawaban yang tepat adalah A.

1

Roboguru

Bentuk setara dengan bentuk ....

Pembahasan Soal:

Perhatikan bahwa

begin mathsize 14px style sum from q equals 6 to 20 of q squared equals 6 squared plus 7 squared plus 8 squared plus 9 squared plus 10 squared plus horizontal ellipsis plus 20 squared equals open square brackets 6 squared plus 7 squared plus 8 squared plus 9 squared close square brackets plus open square brackets 10 squared plus horizontal ellipsis plus 20 squared close square brackets equals sum from q equals 6 to 9 of q squared plus sum from q equals 10 to 20 of q squared end style

Jadi, jawaban yang tepat adalah B.

0

Roboguru

Bentuk  setara dengan bentuk ....

Pembahasan Soal:

Perhatikan penjabaran berikut!

table attributes columnalign right center left columnspacing 0px end attributes row cell sum from k equals 0 to m of U subscript k end cell equals cell U subscript 0 plus U subscript 1 plus U subscript 2 plus... plus U subscript m end cell row cell sum from k equals 0 to m of U subscript k end cell equals cell U subscript 0 plus sum from k equals 1 to m of U subscript k end cell end table

Oleh karena itu, didapat hasil perhitungan sebagai berikut.

sum from k equals 0 to m of open parentheses k minus 5 close parentheses equals open parentheses 0 minus 5 close parentheses plus sum from k equals 1 to m of open parentheses k minus 5 close parentheses equals negative 5 plus open parentheses open parentheses 1 minus 5 close parentheses plus open parentheses 2 minus 5 close parentheses plus... plus open parentheses m minus 5 close parentheses close parentheses equals negative 5 plus open parentheses open parentheses negative 4 close parentheses plus open parentheses negative 3 close parentheses plus... plus open parentheses m minus 5 close parentheses close parentheses equals negative 5 plus sum from k equals negative 4 to m minus 5 of k equals negative 5 plus sum from j equals negative 4 to m minus 5 of j

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Jika bentuk  setara dengan ....

Pembahasan Soal:

Perhatikan penjabaran berikut!

sum from i equals negative 1 to n of open parentheses i plus 2 close parentheses squared equals open parentheses negative 1 plus 2 close parentheses squared plus open parentheses 0 plus 2 close parentheses squared plus open parentheses 1 plus 2 close parentheses squared plus... plus open parentheses n plus 2 close parentheses squared equals 1 squared plus 2 squared plus 3 squared plus... plus open parentheses n plus 2 close parentheses squared equals sum from i equals 1 to n plus 2 of i squared 

Dengan demikian, bentuk sum from i equals negative 1 to n of open parentheses i plus 2 close parentheses squared setara dengan sum from i equals 1 to n plus 2 of i squared.

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Hasil dari adalah ....

Pembahasan Soal:

Kita jabarkan untuk melihat polanya.

Untuk k = 1, maka 2(1 - 1) + 4(1) - 6(1 + 1) = 2(0) + 4(1) - 6(2)

Untuk k = 2, maka 2(2 - 1) + 4(2) - 6(2 + 1) = 2(1) + 4(2) - 6(3)

Untuk k = 3, maka 2(3 - 1) + 4(3) - 6(3 + 1) = 2(2) + 4(3) - 6(4)
.
.
.

Untuk k = 12, maka 2(12 - 1) + 4(12) - 6(12 + 1) = 2(11) + 4(12) - 6(13)

Untuk k = 13, maka 2(13 - 1) + 4(13) - 6(13 + 1) = 2(12) + 4(13) - 6(14)

Untuk k =14, maka 2(14 - 1) + 4(14) - 6(14 + 1) = 2(13) + 4(14) - 6(15)

Jika semua hasil penjabaran di atas dijumlahkan, hasilnya adalah

2(0) + 2(1) + 4(1) + 4(14) - 6(14) - 6(15)

= 2(0) + (2 + 4)(1) + (4 - 6) (14) - 6(15)

= 2(0) + 6(1) - 2(14) - 6(15)

Jadi

begin mathsize 14px style sum from k equals 1 to 14 of open parentheses 2 open parentheses k minus 1 close parentheses plus 4 k minus 6 open parentheses k plus 1 close parentheses close parentheses equals 2 open parentheses 0 close parentheses plus 6 open parentheses 1 close parentheses minus 2 open parentheses 14 close parentheses minus 6 left parenthesis 15 right parenthesis end style 

Dengan demikian, jawaban yang tepat adalah D.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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Kurang Membantu

Apakah pembahasan ini membantu?

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