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Buktikan dengan prinsip induksi matematika: a. 1 + 3 + 5 + ... + ( 2 n − 1 ) + ( 2 n − 3 ) + ... + 3 + 1 = 2 n 2 − 2 n + 1

Buktikan dengan prinsip induksi matematika: 

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terbukti bahwa karena hasil ruas kanan dan kiri sama

terbukti bahwa 1 plus 3 plus 5 plus... plus open parentheses 2 straight n minus 1 close parentheses plus open parentheses 2 straight n minus 3 close parentheses plus... plus 3 plus 1 equals 2 straight n squared minus 2 straight n plus 1 karena hasil ruas kanan dan kiri sama

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Pembuktian menggunakan induksi matematika Untuk n = 1 Untuk n = k diasumsikan benar Untuk n = k+1 maka Jadi terbukti bahwa karena hasil ruas kanan dan kiri sama

Pembuktian menggunakan induksi matematika

Untuk n = 1

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript 1 end cell equals cell 2 straight n squared minus 2 straight n plus 1 end cell row blank equals cell 2.1 squared minus 2.1 plus 1 end cell row blank equals cell 2 minus 2 plus 1 end cell row blank equals cell 1 rightwards arrow terbukti end cell end table

Untuk n = k diasumsikan benar

1 plus 3 plus 5 plus... plus open parentheses 2 straight n minus 1 close parentheses plus open parentheses 2 straight n minus 3 close parentheses plus... plus 3 plus 1 equals 2 straight n squared minus 2 straight n plus 1 1 plus 3 plus 5 plus... plus open parentheses 2 straight k minus 1 close parentheses plus open parentheses 2 straight k minus 3 close parentheses plus... plus 3 plus 1 equals 2 straight k squared minus 2 straight k plus 1 rightwards arrow diasumsikan space benar

Untuk n = k+1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 plus 3 plus 5 plus... plus open parentheses 2 straight n minus 1 close parentheses plus open parentheses 2 straight n minus 3 close parentheses plus... plus 3 plus 1 end cell equals cell 2 straight n squared minus 2 straight n plus 1 end cell row cell 1 plus 3 plus 5 plus... plus open parentheses 2 left parenthesis straight k plus 1 right parenthesis minus 1 close parentheses plus open parentheses 2 left parenthesis straight k plus 1 right parenthesis minus 3 close parentheses plus... plus 3 plus 1 end cell equals cell 2 left parenthesis straight k plus 1 right parenthesis squared minus 2 left parenthesis straight k plus 1 right parenthesis plus 1 end cell row cell 1 plus 3 plus 5 plus... plus open parentheses 2 straight k plus 2 minus 1 close parentheses plus open parentheses 2 straight k plus 2 minus 3 close parentheses plus... plus 3 plus 1 end cell equals cell 2 left parenthesis straight k squared plus 2 straight k plus 1 right parenthesis minus 2 straight k minus 2 plus 1 end cell row cell 1 plus 3 plus 5 plus... plus open parentheses 2 straight k plus 1 close parentheses plus open parentheses 2 straight k minus 1 close parentheses plus... plus 3 plus 1 end cell equals cell 2 straight k squared plus 4 straight k plus 2 minus 2 straight k minus 2 plus 1 end cell row cell 1 plus 3 plus 5 plus... plus open parentheses 2 straight k plus 1 close parentheses plus open parentheses 2 straight k minus 1 close parentheses plus... plus 3 plus 1 end cell equals cell 2 straight k squared plus 2 straight k plus 1 end cell row cell 1 plus 3 plus 5 plus... plus open parentheses 2 straight k minus 1 close parentheses plus open parentheses 2 straight k plus 1 close parentheses plus open parentheses 2 straight k minus 1 close parentheses plus... plus 3 plus 1 end cell equals cell 2 straight k squared plus 2 straight k plus 1 end cell row cell 2 straight k squared minus 2 straight k plus 1 plus open parentheses 2 straight k plus 1 close parentheses plus open parentheses 2 straight k minus 1 close parentheses end cell equals cell 2 straight k squared plus 2 straight k plus 1 end cell row cell 2 straight k squared up diagonal strike negative 2 straight k end strike plus 1 up diagonal strike plus 2 straight k end strike up diagonal strike plus 1 end strike plus 2 straight k up diagonal strike negative 1 end strike end cell equals cell 2 straight k squared plus 2 straight k plus 1 end cell row cell 2 straight k squared plus 2 straight k plus 1 end cell equals cell 2 straight k squared plus 2 straight k plus 1 rightwards arrow terbukti end cell end table

Jadi terbukti bahwa 1 plus 3 plus 5 plus... plus open parentheses 2 straight n minus 1 close parentheses plus open parentheses 2 straight n minus 3 close parentheses plus... plus 3 plus 1 equals 2 straight n squared minus 2 straight n plus 1 karena hasil ruas kanan dan kiri sama

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