Roboguru

Buktikan bahwa:  a.  Pn​≡1×31​+3×51​+...+(2n−1)(2n+1)1​=2n+1n​

Pertanyaan

Buktikan bahwa: 

a.  straight P subscript straight n identical to fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2 straight n minus 1 close parentheses open parentheses 2 straight n plus 1 close parentheses end fraction equals fraction numerator straight n over denominator 2 straight n plus 1 end fraction   

Pembahasan Soal:

Membuktikan dengan induksi matematika dimana

Untuk n = 1

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript straight n end cell identical to cell fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2 straight n minus 1 close parentheses open parentheses 2 straight n plus 1 close parentheses end fraction equals fraction numerator straight n over denominator 2 straight n plus 1 end fraction end cell row blank blank cell space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 1 third equals 1 third rightwards arrow terbukti end cell row blank blank blank end table

Untuk n = k diasumsikan terbukti maka

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript straight n end cell identical to cell fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2 straight n minus 1 close parentheses open parentheses 2 straight n plus 1 close parentheses end fraction equals fraction numerator straight n over denominator 2 straight n plus 1 end fraction end cell row cell straight P subscript straight k end cell identical to cell fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2. straight k minus 1 close parentheses open parentheses 2. straight k plus 1 close parentheses end fraction equals fraction numerator straight k over denominator 2. straight k plus 1 end fraction end cell row blank blank blank end table

Untuk n = k+1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P subscript straight n end cell identical to cell fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2 straight n minus 1 close parentheses open parentheses 2 straight n plus 1 close parentheses end fraction equals fraction numerator straight n over denominator 2 straight n plus 1 end fraction end cell row cell straight P subscript straight k plus 1 end subscript end cell identical to cell fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2 straight k minus 1 close parentheses open parentheses 2 straight k plus 1 close parentheses end fraction plus fraction numerator 1 over denominator open parentheses 2 straight k plus 1 close parentheses open parentheses 2 straight k plus 3 close parentheses end fraction equals fraction numerator straight k plus 1 over denominator 2 straight k plus 3 end fraction rightwards arrow terbukti end cell row blank blank blank end table

Akan dibuktikan bahwa

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

Jadi terbukti straight P subscript straight n identical to fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 5 end fraction plus... plus fraction numerator 1 over denominator open parentheses 2 straight n minus 1 close parentheses open parentheses 2 straight n plus 1 close parentheses end fraction equals fraction numerator straight n over denominator 2 straight n plus 1 end fraction karena hasil ruas kiri dan kanan sama

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 07 Oktober 2021

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