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Buktikan dengan prinsip induksi matematika bahwa semua bilangan asli n selalu berlaku: P n ​ ≡ ( n + 1 ) 2 < n 3 , untuk n ≥ 3

Buktikan dengan prinsip induksi matematika bahwa semua bilangan asli n selalu berlaku: 

, untuk     

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A. Acfreelance

Master Teacher

Mahasiswa/Alumni UIN Walisongo Semarang

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terbukti bahwa karena

terbukti bahwa straight P subscript straight n identical to left parenthesis straight n plus 1 right parenthesis squared less than straight n cubed karena table attributes columnalign right center left columnspacing 0px end attributes row cell straight k cubed plus straight k left parenthesis straight k plus 4 right parenthesis plus 4 end cell less than cell straight k cubed plus 3 straight k left parenthesis straight k plus 1 right parenthesis plus 1 end cell end table

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Pembuktian dengan menggunakan induksi matematika Untuk n = 3 maka Untuk n = k diasumsikan terbukti Untukn = k+1 maka Jadi terbukti bahwa karena

Pembuktian dengan menggunakan induksi matematika

Untuk n = 3 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis straight n plus 1 right parenthesis squared end cell less than cell straight n cubed end cell row cell open parentheses 3 plus 1 close parentheses squared end cell less than cell 3 cubed end cell row cell 4 squared end cell less than cell 3 cubed end cell row 16 less than cell 27 rightwards arrow Terbukti end cell end table

Untuk n = k diasumsikan terbukti

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis straight n plus 1 right parenthesis squared end cell less than cell straight n cubed end cell row cell open parentheses straight k plus 1 close parentheses squared end cell less than cell straight k cubed rightwards arrow Terbukti end cell end table

Untukn = k+1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis straight n plus 1 right parenthesis squared end cell less than cell straight n cubed end cell row cell open parentheses straight k plus 1 plus 1 close parentheses squared end cell less than cell left parenthesis straight k plus 1 right parenthesis cubed space end cell row cell straight k cubed plus left parenthesis straight k plus 2 right parenthesis squared end cell less than cell left parenthesis straight k squared plus 2 straight k plus 1 right parenthesis left parenthesis straight k plus 1 right parenthesis end cell row cell straight k cubed plus straight k squared plus 4 straight k plus 4 end cell less than cell straight k cubed plus straight k squared plus 2 straight k squared plus 2 straight k plus straight k plus 1 end cell row cell straight k cubed plus straight k squared plus 4 straight k plus 4 end cell less than cell straight k cubed plus 3 straight k squared plus 3 straight k plus 1 end cell row cell straight k cubed plus straight k left parenthesis straight k plus 4 right parenthesis plus 4 end cell less than cell straight k cubed plus 3 straight k left parenthesis straight k plus 1 right parenthesis plus 1 rightwards arrow Terbukti end cell end table

Jadi terbukti bahwa straight P subscript straight n identical to left parenthesis straight n plus 1 right parenthesis squared less than straight n cubed karena table attributes columnalign right center left columnspacing 0px end attributes row cell straight k cubed plus straight k left parenthesis straight k plus 4 right parenthesis plus 4 end cell less than cell straight k cubed plus 3 straight k left parenthesis straight k plus 1 right parenthesis plus 1 end cell end table

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Nasywa Qothrunnada Kamilah

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