Roboguru

Tunjukkan bahwa untuk semua bilangan bulat positif  selalu berlaku:  a.

Pertanyaan

Tunjukkan bahwa untuk semua bilangan bulat positif undefined selalu berlaku: 

a. 1 plus 2 straight x plus 3 straight x squared plus... plus nx to the power of straight n minus 1 end exponent equals fraction numerator 1 minus straight x to the power of straight n over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator nx to the power of straight n over denominator 1 minus straight x end fraction 

Pembahasan Soal:

Untuk membuktikan bahwa bilangan bulat positif berlaku pada pembuktian di bawah kita misalkan bahwa n = 1 maka didapat

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 plus 2 straight x plus 3 straight x squared plus... plus nx to the power of straight n minus 1 end exponent end cell equals cell fraction numerator 1 minus straight x to the power of straight n over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator nx to the power of straight n over denominator 1 minus straight x end fraction end cell row cell 1 plus 2 straight x plus 3 straight x squared plus... plus 1. straight x to the power of 1 minus 1 end exponent end cell equals cell fraction numerator 1 minus straight x to the power of 1 over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator 1. straight x to the power of 1 over denominator 1 minus straight x end fraction end cell row cell 1. straight x to the power of 0 end cell equals cell fraction numerator 1 minus straight x over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator 1. straight x over denominator 1 minus straight x end fraction end cell row 1 equals cell fraction numerator 1 minus straight x minus open parentheses 1 minus straight x close parentheses straight x over denominator open parentheses 1 minus straight x close parentheses squared end fraction end cell row 1 equals cell 1 rightwards arrow terbukti end cell row blank blank blank end table

Sedangkan untuk n = -1 maka di dapat

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 plus 2 straight x plus 3 straight x squared plus... plus nx to the power of straight n minus 1 end exponent end cell equals cell fraction numerator 1 minus straight x to the power of straight n over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator nx to the power of straight n over denominator 1 minus straight x end fraction end cell row cell 1 plus 2 straight x plus 3 straight x squared plus... plus negative 1. straight x to the power of negative 1 minus 1 end exponent end cell equals cell fraction numerator 1 minus straight x to the power of negative 1 end exponent over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator negative 1. straight x to the power of negative 1 end exponent over denominator 1 minus straight x end fraction end cell row cell negative straight x to the power of negative 2 end exponent end cell equals cell fraction numerator 1 minus begin display style 1 over straight x end style over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator negative straight x to the power of negative 1 end exponent over denominator 1 minus straight x end fraction end cell row cell negative 1 over straight x squared end cell equals cell fraction numerator begin display style fraction numerator straight x minus 1 over denominator straight x end fraction end style over denominator open parentheses 1 minus straight x close parentheses squared end fraction plus fraction numerator begin display style 1 over straight x end style over denominator 1 minus straight x end fraction end cell row cell negative 1 over straight x squared end cell equals cell fraction numerator straight x minus 1 over denominator straight x end fraction cross times 1 over open parentheses 1 minus straight x close parentheses squared plus 1 over straight x cross times fraction numerator 1 over denominator 1 minus straight x end fraction end cell row cell negative 1 over straight x squared end cell equals cell fraction numerator open parentheses straight x minus 1 close parentheses 1 plus straight x minus 1 over denominator straight x open parentheses 1 minus straight x close parentheses squared end fraction end cell row cell negative 1 over straight x squared end cell equals cell fraction numerator straight x minus 1 plus straight x minus 1 over denominator straight x open parentheses 1 minus straight x close parentheses squared end fraction end cell row cell negative 1 over straight x squared end cell equals cell fraction numerator 2 straight x minus 2 over denominator straight x open parentheses 1 minus straight x close parentheses squared end fraction rightwards arrow tidak space terbukti end cell row blank blank blank end table

Jadi karena dalam pembuktian n = -1 tidak terbukti karena hasil kanan dan kiri tidak sama maka untuk n bilan bulat positif terbukti untuk 1 plus 2 straight x plus 3 straight x squared plus... plus nx to the power of straight n minus 1 end exponent equals fraction numerator 1 minus straight x to the power of straight n over denominator open parentheses 1 minus straight x close parentheses squared end fraction minus fraction numerator nx to the power of straight n over denominator 1 minus straight x end fraction

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 18 Juli 2021

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