Pertanyaan

Buktikan dengan induksi matematika bahwa rumus berikut berlaku untuk setiap bilangan asli! b.

Buktikan dengan induksi matematika bahwa rumus berikut berlaku untuk setiap bilangan asli!

b. $begin mathsize 14px style 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus n over 2 to the power of n equals 2 minus fraction numerator n plus 2 over denominator 2 to the power of n end fraction end style$

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rumus berlaku untuk setiap bilangan asli.

rumus $begin mathsize 14px style 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus n over 2 to the power of n equals 2 minus fraction numerator n plus 2 over denominator 2 to the power of n end fraction end style$ berlaku untuk setiap

bilangan asli.

Pembahasan

Perhatikan perhitungan berikut ini. 1. Akan dibuktikan untuk adalah benar, maka: Benar untuk 2. Asumsikan , maka: Akan dibuktikan benar, sehingga: Pembuktian: Terbukti. Jadi,rumus berlaku untuk setiap bilangan asli.

Perhatikan perhitungan berikut ini.

$begin mathsize 14px style 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus n over 2 to the power of n equals 2 minus fraction numerator n plus 2 over denominator 2 to the power of n end fraction end style$

1. Akan dibuktikan untuk adalah benar, maka:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 half end cell equals cell 2 minus fraction numerator n plus 2 over denominator 2 to the power of n end fraction end cell row cell 1 half end cell equals cell 2 minus fraction numerator 1 plus 2 over denominator 2 to the power of 1 end fraction end cell row cell 1 half end cell equals cell 2 minus 3 over 2 end cell row cell 1 half end cell equals cell fraction numerator 4 minus 3 over denominator 2 end fraction end cell row cell 1 half end cell equals cell 1 half end cell end table end style$

Benar untuk

2. Asumsikan , maka:

$begin mathsize 14px style 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus k over 2 to the power of k equals 2 minus fraction numerator k plus 2 over denominator 2 to the power of k end fraction end style$

Akan dibuktikan benar, sehingga:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus k over 2 to the power of k plus fraction numerator k plus 1 over denominator 2 to the power of k plus 1 end exponent end fraction end cell equals cell 2 minus fraction numerator open parentheses k plus 1 close parentheses plus 2 over denominator 2 to the power of k plus 1 end exponent end fraction end cell end table end style$

Pembuktian:

$begin mathsize 14px style 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus k over 2 to the power of k plus fraction numerator k plus 1 over denominator 2 to the power of k plus 1 end exponent end fraction space equals 2 minus fraction numerator k plus 2 over denominator 2 to the power of k end fraction plus fraction numerator k plus 1 over denominator 2 to the power of k plus 1 end exponent end fraction space equals 2 minus open parentheses fraction numerator k plus 2 over denominator 2 to the power of k end fraction minus fraction numerator k plus 1 over denominator 2 to the power of k plus 1 end exponent end fraction space close parentheses equals 2 minus open parentheses fraction numerator 2 open parentheses k plus 2 close parentheses over denominator 2 times 2 to the power of k end fraction minus fraction numerator k plus 1 over denominator 2 to the power of k plus 1 end exponent end fraction close parentheses space equals 2 minus open parentheses fraction numerator 2 k plus 4 over denominator 2 to the power of k plus 1 end exponent end fraction minus fraction numerator k plus 1 over denominator 2 to the power of k plus 1 end exponent end fraction space close parentheses equals 2 minus open parentheses fraction numerator 2 k plus 4 minus open parentheses k plus 1 close parentheses over denominator 2 to the power of k plus 1 end exponent end fraction close parentheses space equals 2 minus fraction numerator 2 k plus 4 minus k minus 1 over denominator 2 to the power of k plus 1 end exponent end fraction space equals 2 minus fraction numerator k plus 3 over denominator 2 to the power of k plus 1 end exponent end fraction space equals 2 minus fraction numerator open parentheses k plus 1 close parentheses plus 2 over denominator 2 to the power of k plus 1 end exponent end fraction end style$

Terbukti.

Jadi, rumus $begin mathsize 14px style 1 half plus 2 over 4 plus 3 over 8 plus 4 over 16 plus... plus n over 2 to the power of n equals 2 minus fraction numerator n plus 2 over denominator 2 to the power of n end fraction end style$ berlaku untuk setiap bilangan asli.

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