Pertanyaan

Hitunglah nilai dari ( 1 − 2 2 1 ) ( 1 − 3 2 1 ) ( 1 − 4 2 1 ) ... ( 1 − 9 9 2 1 ) !

Hitunglah nilai dari $(1 - \frac{1}{2 ^{2}}) (1 - \frac{1}{3 ^{2}}) (1 - \frac{1}{4 ^{2}}) ... (1 - \frac{1}{9 9 ^{2}})$ !

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Jawaban terverifikasi

Pembahasan

Perhatikan bahwa maka diperoleh Selanjutnya,nilai sebagai berikut. Jadi, .

Perhatikan bahwa $begin mathsize 14px style 1 minus 1 over n squared equals fraction numerator n squared minus 1 over denominator n to the power of 2 space end exponent end fraction equals fraction numerator open parentheses n minus 1 close parentheses open parentheses n plus 1 close parentheses over denominator n squared end fraction end style$ maka diperoleh

$begin mathsize 14px style 1 minus 1 over 2 squared equals fraction numerator open parentheses 2 minus 1 close parentheses open parentheses 2 plus 1 close parentheses over denominator 2 squared end fraction equals fraction numerator 1 times 3 over denominator 2 squared end fraction end style$

$begin mathsize 14px style 1 minus 1 over 3 squared equals fraction numerator open parentheses 3 minus 1 close parentheses open parentheses 3 plus 1 close parentheses over denominator 3 squared end fraction equals fraction numerator 2 times 4 over denominator 3 squared end fraction end style$

$begin mathsize 14px style 1 minus 1 over 4 squared equals fraction numerator open parentheses 4 minus 1 close parentheses open parentheses 4 plus 1 close parentheses over denominator 4 squared end fraction equals fraction numerator 3 times 5 over denominator 4 squared end fraction end style$

begin mathsize 14px style vertical ellipsis end style

$begin mathsize 14px style 1 minus 1 over 99 squared equals fraction numerator open parentheses 99 minus 1 close parentheses open parentheses 99 plus 1 close parentheses over denominator 99 squared end fraction equals fraction numerator 98 times 100 over denominator 99 squared end fraction end style$

Selanjutnya, nilai begin mathsize 14px style open parentheses 1 minus 1 over 2 squared close parentheses open parentheses 1 minus 1 over 3 squared close parentheses open parentheses 1 minus 1 over 4 squared close parentheses... open parentheses 1 minus 1 over 99 squared close parentheses end style sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 minus 1 over 2 squared close parentheses open parentheses 1 minus 1 over 3 squared close parentheses open parentheses 1 minus 1 over 4 squared close parentheses midline horizontal ellipsis open parentheses 1 minus 1 over 99 squared close parentheses end cell row blank equals cell open parentheses fraction numerator 1 times 3 over denominator 2 squared end fraction close parentheses open parentheses fraction numerator 2 times 4 over denominator 3 squared end fraction close parentheses open parentheses fraction numerator 3 times 5 over denominator 4 squared end fraction close parentheses midline horizontal ellipsis open parentheses fraction numerator 98 times 100 over denominator 99 squared end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator 1 times 3 over denominator 2 to the power of up diagonal strike 2 end exponent end fraction close parentheses open parentheses fraction numerator up diagonal strike 2 times 4 over denominator 3 to the power of up diagonal strike 2 end exponent end fraction close parentheses open parentheses fraction numerator up diagonal strike 3 times 5 over denominator 4 to the power of up diagonal strike 2 end exponent end fraction close parentheses midline horizontal ellipsis open parentheses fraction numerator up diagonal strike 98 times 100 over denominator 99 squared end fraction close parentheses end cell row blank equals cell open parentheses 3 over 2 close parentheses open parentheses 4 over 3 close parentheses open parentheses 5 over 4 close parentheses midline horizontal ellipsis open parentheses 100 over 99 squared close parentheses end cell row blank equals cell open parentheses fraction numerator up diagonal strike 3 over denominator 2 end fraction close parentheses open parentheses fraction numerator up diagonal strike 4 over denominator up diagonal strike 3 end fraction close parentheses open parentheses fraction numerator up diagonal strike 5 over denominator up diagonal strike 4 end fraction close parentheses midline horizontal ellipsis open parentheses 100 over 99 to the power of up diagonal strike 2 end exponent close parentheses end cell row blank equals cell fraction numerator 100 over denominator 2 times 99 end fraction end cell row blank equals cell 100 over 198 end cell end table end style$

Jadi, begin mathsize 14px style open parentheses 1 minus 1 over 2 squared close parentheses open parentheses 1 minus 1 over 3 squared close parentheses open parentheses 1 minus 1 over 4 squared close parentheses... open parentheses 1 minus 1 over 99 squared close parentheses equals 100 over 198 end style .