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Buktikan dengan prinsip induksi matematika bahwa semua bilangan asli n selalu berlaku:  Pn​≡(n+1)2<2n, untuk n≥6

Pertanyaan

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

straight P subscript straight n identical to left parenthesis straight n plus 1 right parenthesis squared less than 2 to the power of straight n, untuk straight n greater or equal than 6 

A. Acfreelance

Master Teacher

Mahasiswa/Alumni UIN Walisongo Semarang

Jawaban terverifikasi

Jawaban

terbukti bahwa straight P subscript straight n identical to left parenthesis straight n plus 1 right parenthesis squared less than 2 to the power of straight n dimana straight n greater or equal than 6 karena hasil table attributes columnalign right center left columnspacing 0px end attributes row cell 2 to the power of straight k plus straight k squared plus 4 open parentheses straight k plus 1 close parentheses end cell less than cell 2 to the power of straight k plus 1 end exponent end cell end table

Pembahasan

Pembuktian dengan induksi matematika dimana

n = 6maka

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 2 to the power of straight n end cell row cell left parenthesis 6 plus 1 right parenthesis squared end cell less than cell 2 to the power of 6 end cell row cell 7 squared end cell less than cell 2 to the power of 6 end cell row 49 less than cell 64 rightwards arrow Terbukti end cell end table

Untuk n = k diasumsikan terbukti 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 2 to the power of straight n end cell row cell left parenthesis straight k plus 1 right parenthesis squared end cell less than cell 2 to the power of straight k rightwards arrow Terbukti end cell end table

Untuk n = k+1 maka akan dibuktikan

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 2 to the power of straight n end cell row cell left parenthesis straight k plus 1 right parenthesis squared plus left parenthesis straight k plus 2 right parenthesis squared end cell less than cell 2 to the power of straight k plus 1 end exponent space end cell row cell 2 to the power of straight k plus straight k squared plus 4 straight k plus 4 end cell less than cell 2 to the power of straight k plus 1 end exponent end cell row cell 2 to the power of straight k plus straight k squared plus 4 open parentheses straight k plus 1 close parentheses end cell less than cell 2 to the power of straight k plus 1 end exponent 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 2 to the power of straight n dimana straight n greater or equal than 6 karena hasil table attributes columnalign right center left columnspacing 0px end attributes row cell 2 to the power of straight k plus straight k squared plus 4 open parentheses straight k plus 1 close parentheses end cell less than cell 2 to the power of straight k plus 1 end exponent end cell end table

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