Pertanyaan

Buktikan masing-masing notasi sigma berikut. b. j = 1 ∑ n ( 3 j − 2 ) = 2 3 n 2 − n

Buktikan masing-masing notasi sigma berikut.

b. $j = 1 \sum n (3 j - 2) = \frac{3 n ^{2} - n}{2}$

A. Acfreelance

Master Teacher

Mahasiswa/Alumni UIN Walisongo Semarang

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Pembahasan

Dibuktikan menggunakan induksi matematika dimana Untuk n = 1 Untuk n = k dimana diasumsikan terbukti Untuk n = k+1 maka PRK Jadi dengan menggunakan induksi matematika terbukti karena hasil dari sisi kanan dan kiri sama

Dibuktikan menggunakan induksi matematika dimana

Untuk n = 1

Untuk n = k dimana diasumsikan terbukti

$table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight j equals 1 to straight n of open parentheses 3 straight j minus 2 close parentheses end cell equals cell fraction numerator 3 straight n squared minus straight n over denominator 2 end fraction end cell row cell sum from straight j equals 1 to straight k of open parentheses 3 straight j minus 2 close parentheses end cell equals cell fraction numerator 3. straight k squared minus straight k over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 straight k squared minus straight k over denominator 2 end fraction rightwards arrow terbukti end cell row blank blank blank end table$

Untuk n = k+1 maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight j equals 1 to straight n of open parentheses 3 straight j minus 2 close parentheses end cell equals cell fraction numerator 3 straight n squared minus straight n over denominator 2 end fraction end cell row cell sum from straight j equals 1 to straight k plus 1 of open parentheses 3 straight j minus 2 close parentheses end cell equals cell fraction numerator 3 left parenthesis straight k plus 1 right parenthesis squared minus left parenthesis straight k plus 1 right parenthesis over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 left parenthesis straight k squared plus 2 straight k plus 1 right parenthesis minus straight k minus 1 over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 straight k squared plus 5 straight k plus 2 over denominator 2 end fraction rightwards arrow terbukti end cell row blank blank blank end table$

PRK

$table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight j equals 1 to straight k of open parentheses 3 straight j minus 2 close parentheses end cell equals cell sum from straight j equals 1 to k of open parentheses 3 straight j minus 2 close parentheses plus sum from straight j equals k plus 1 to straight k plus 1 of open parentheses 3 straight j minus 2 close parentheses end cell row cell fraction numerator 3 straight k squared plus 5 straight k plus 2 over denominator 2 end fraction end cell equals cell fraction numerator 3 straight k squared minus straight k over denominator 2 end fraction plus 3 open parentheses k plus 1 close parentheses minus 2 end cell row cell fraction numerator 3 straight k squared plus 5 straight k plus 2 over denominator 2 end fraction end cell equals cell fraction numerator 3 straight k squared minus straight k over denominator 2 end fraction plus fraction numerator 6 straight k plus 2 over denominator 2 end fraction end cell row cell fraction numerator 3 straight k squared plus 5 straight k plus 2 over denominator 2 end fraction end cell equals cell fraction numerator 3 straight k squared plus 5 straight k plus 2 over denominator 2 end fraction rightwards arrow terbukti end cell row blank blank blank end table$

Jadi $sum from straight j equals 1 to straight n of open parentheses 3 straight j minus 2 close parentheses equals fraction numerator 3 straight n squared minus straight n over denominator 2 end fraction$ dengan menggunakan induksi matematika terbukti karena hasil dari sisi kanan dan kiri sama

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