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Petunjuk:Pergunakan prinsip induksi matematika untuk membuktikan setiap pernyataan berikut. 1. 2 + 4 + 6 + ... + ( 2 n ) = i = 1 ∑ n ​ ( 2 i ) = n ( n + 1 )

Petunjuk: Pergunakan prinsip induksi matematika untuk membuktikan setiap pernyataan berikut. 

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Langkah Pertama Misalkan n=1 maka Langkah kedua Anggaplah n=k benar maka Akan dibuktikan bahwa Perhatikan PRK : Jadi terbukti bahwa karena ruas kanan dan kiri mempunyai nilai sama yaitu

Langkah Pertama

Misalkan n=1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript 1 end cell identical to cell sum from straight i equals 1 to straight n of open parentheses 2 straight i close parentheses equals straight n left parenthesis straight n plus 1 right parenthesis end cell row blank identical to cell sum from straight i equals 1 to straight n of left parenthesis 2.1 right parenthesis equals 1 left parenthesis 1 plus 1 right parenthesis end cell row blank blank cell space space space space space space space space space 2 equals 2 space rightwards arrow space terbukti space end cell end table

Langkah kedua

Anggaplah n=k benar maka

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript straight k end cell identical to cell sum from straight i equals 1 to straight n of open parentheses 2 straight i close parentheses equals straight n left parenthesis straight n plus 1 right parenthesis end cell row blank identical to cell sum from straight i equals 1 to straight k of left parenthesis 2 straight i right parenthesis equals straight k left parenthesis straight k plus 1 right parenthesis rightwards arrow Anggap space benar end cell end table

Akan dibuktikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript straight k plus 1 end subscript end cell identical to cell sum from straight i equals 1 to straight n of open parentheses 2 straight i close parentheses equals straight n left parenthesis straight n plus 1 right parenthesis end cell row blank identical to cell sum from straight i equals 1 to straight k plus 1 of left parenthesis 2 straight i right parenthesis equals straight k plus 1 left parenthesis straight k plus 1 plus 1 right parenthesis end cell row blank blank cell space space space space space space space space space space space equals left parenthesis straight k plus 1 right parenthesis left parenthesis straight k plus 2 right parenthesis equals straight k squared plus 2 straight k plus straight k plus 2 equals straight k squared plus 3 straight k plus 2 end cell end table

Perhatikan PRK :

sum from straight i equals 1 to straight k plus 1 of open parentheses 2 straight i close parentheses space space space space space equals sum from straight i equals 1 to straight k of left parenthesis 2 straight i right parenthesis plus sum from straight i equals straight k plus 1 to straight k plus 1 of left parenthesis 2 straight i right parenthesis straight k squared plus 3 straight k plus 2 equals straight k left parenthesis straight k plus 1 right parenthesis plus 2 left parenthesis straight k plus 1 right parenthesis straight k squared plus 3 straight k plus 2 equals straight k squared plus 3 straight k plus 2

Jadi terbukti bahwa 2 plus 4 plus 6 plus... plus open parentheses 2 straight n close parentheses equals sum from straight i equals 1 to straight n of open parentheses 2 straight i close parentheses equals straight n open parentheses straight n plus 1 close parentheses karena ruas kanan dan kiri mempunyai nilai sama yaitu straight k squared plus 3 straight k plus 2

 

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