Seperti naik turun tangga

Posted on October 11, 2013 by Nursatria
Sumber: Self.com

Sumber: Self.com

Coba perhatikan:

1+2+1=4=2^2

1+2+3+2+1=9=3^2

1+2+3+4+3+2+1=16=4^2

1+2+3+4+5+4+3+2+1=25=5^2

Sudah terlihat polanya kan? Deret-deret diatas seperti naik turun tannga, bermula dari 1 naik selangkah demi selangkah sampai ke puncak lalu turun kembali.

Sekarang mari kita buktikan bentuk umumnya:

1+2+3+4+\ldots+\left(n-1\right)+n+\left(n-1\right)+\ldots+4+3+2+1=n^{2}

Bukti:

1+2+3+4+\ldots+\left(n-1\right)+n+\left(n-1\right)+\ldots+4+3+2+1

=1+2+3+4+\ldots+\left(n-1\right)+1+2+3+4+\ldots+\left(n-1\right)+n

=2\left(1+2+3+4+\ldots+\left(n-1\right)\right)+n

=2\left(\frac{\left(n-1\right)n}{2}\right)+n

=\left(n-1\right)n+n

=n^2-n+n

=n^2

\square

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About Nursatria

Seorang Alumnus Matematika UGM, dengan ilmu yang didapat ketika kuliah (Padahal sering bolos kuliah :p ), saya menyebarkan virus matematika
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3 Responses to Seperti naik turun tangga

  1. Sungkono says:
    October 12, 2013 at 22:10

    Tertulis : 1+2+3+4+…+(n+1)+n+(n+1)+…4+3+2+1 sedikit mengganggu, mungkin seharusnya: 1+2+3+4+…+(n-1)+n+(n-1)+…4+3+2+1

    Reply
  2. Nene says:
    October 11, 2013 at 22:28

    lagi nyari ide buat penelitian matematika kak. tolong bantu ya !!!

    Reply

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