Pertanyaan

Dengan menggunakan prinsip induksi matematika, buktikanlah: c. P n ≡ 1.3 1 + 2.4 1 + 3.5 1 + ... + n ( n + 2 ) 1 = 4 ( n + 1 ) ( n + 2 ) n ( 3 n + 5 )

Dengan menggunakan prinsip induksi matematika, buktikanlah:

c. $P_{n} \equiv \frac{1}{1.3} + \frac{1}{2.4} + \frac{1}{3.5} + ... + \frac{1}{n ( n + 2 )} = \frac{n ( 3 n + 5 )}{4 ( n + 1 ) ( n + 2 )}$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

terbukti bahwa benar untuk setiap bilangan asli n .

terbukti bahwa $straight P subscript straight n identical to fraction numerator 1 over denominator 1.3 end fraction plus fraction numerator 1 over denominator 2.4 end fraction plus fraction numerator 1 over denominator 3.5 end fraction plus... plus fraction numerator 1 over denominator straight n open parentheses straight n plus 2 close parentheses end fraction equals fraction numerator straight n open parentheses 3 straight n plus 5 close parentheses over denominator 4 open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses end fraction$ benar untuk setiap bilangan asli

n

Pembahasan

Menggunakan pembuktian induksi matematika dimana untuk n = 1 maka Untuk n = k diasumsikan terbukti maka Untuk n = k+1 maka Jadi terbukti bahwa benar untuk setiap bilangan asli n .

Menggunakan pembuktian induksi matematika dimana

untuk n = 1 maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator straight n open parentheses straight n plus 2 close parentheses end fraction end cell equals cell fraction numerator straight n open parentheses 3 straight n plus 5 close parentheses over denominator 4 open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses end fraction end cell row cell fraction numerator 1 over denominator 1 open parentheses 1 plus 2 close parentheses end fraction end cell equals cell fraction numerator 1 open parentheses 3.1 plus 5 close parentheses over denominator 4 open parentheses 1 plus 1 close parentheses open parentheses 1 plus 2 close parentheses end fraction end cell row cell fraction numerator 1 over denominator 1.3 end fraction end cell equals cell fraction numerator 1.8 over denominator 4 open parentheses 2 close parentheses open parentheses 3 close parentheses end fraction end cell row cell 1 third end cell equals cell 1 third rightwards arrow Terbukti space end cell end table$

Untuk n = k diasumsikan terbukti maka

Untuk n = k+1 maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator 1.3 end fraction plus fraction numerator 1 over denominator 2.4 end fraction plus fraction numerator 1 over denominator 3.5 end fraction plus... plus fraction numerator 1 over denominator straight n open parentheses straight n plus 2 close parentheses end fraction end cell equals cell fraction numerator straight n open parentheses 3 straight n plus 5 close parentheses over denominator 4 open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses end fraction end cell row cell fraction numerator 1 over denominator 1.3 end fraction plus fraction numerator 1 over denominator 2.4 end fraction plus fraction numerator 1 over denominator 3.5 end fraction plus... plus fraction numerator 1 over denominator straight k open parentheses straight k plus 2 close parentheses end fraction plus fraction numerator 1 over denominator left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 1 plus 2 close parentheses end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 left parenthesis straight k plus 1 right parenthesis plus 5 close parentheses over denominator 4 open parentheses straight k plus 1 plus 1 close parentheses open parentheses straight k plus 1 plus 2 close parentheses end fraction end cell row cell fraction numerator k left parenthesis 3 k plus 5 right parenthesis over denominator 4 left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 2 close parentheses end fraction plus fraction numerator 1 over denominator left parenthesis k plus 1 right parenthesis left parenthesis k plus 3 right parenthesis end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell row cell fraction numerator left parenthesis k plus 3 right parenthesis k left parenthesis 3 k plus 5 right parenthesis plus 4 left parenthesis k plus 2 right parenthesis over denominator 4 left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 2 close parentheses left parenthesis k plus 3 right parenthesis end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell row cell fraction numerator left parenthesis k squared plus 3 k right parenthesis left parenthesis 3 k plus 5 right parenthesis plus left parenthesis 4 k plus 8 right parenthesis over denominator 4 left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 2 close parentheses left parenthesis k plus 3 right parenthesis end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell row cell fraction numerator 3 k cubed plus 5 k squared plus 9 k squared plus 15 k plus 4 k plus 8 over denominator 4 left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 2 close parentheses left parenthesis k plus 3 right parenthesis end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell row cell fraction numerator 3 k cubed plus 14 k squared plus 19 k plus 8 over denominator 4 left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 2 close parentheses left parenthesis k plus 3 right parenthesis end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell row cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses left parenthesis k plus 1 right parenthesis over denominator 4 left parenthesis straight k plus 1 right parenthesis open parentheses straight k plus 2 close parentheses left parenthesis k plus 3 right parenthesis end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell row cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction end cell equals cell fraction numerator left parenthesis straight k plus 1 right parenthesis open parentheses 3 straight k plus 8 close parentheses over denominator 4 open parentheses straight k plus 2 close parentheses open parentheses straight k plus 3 close parentheses end fraction rightwards arrow Terbukti space end cell end table$

Jadi terbukti bahwa $straight P subscript straight n identical to fraction numerator 1 over denominator 1.3 end fraction plus fraction numerator 1 over denominator 2.4 end fraction plus fraction numerator 1 over denominator 3.5 end fraction plus... plus fraction numerator 1 over denominator straight n open parentheses straight n plus 2 close parentheses end fraction equals fraction numerator straight n open parentheses 3 straight n plus 5 close parentheses over denominator 4 open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses end fraction$ benar untuk setiap bilangan asli $n$ .

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