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Suku ketiga dari ekspansi  adalah dua kali suku keempatnya. Hitunglah nilai  kemudian tentukan suku tengah dari ekspansi tersebut.

Pertanyaan

Suku ketiga dari ekspansi begin mathsize 14px style open parentheses 1 plus 1 fourth close parentheses to the power of n end style adalah dua kali suku keempatnya. Hitunglah nilai begin mathsize 14px style straight n end style kemudian tentukan suku tengah dari ekspansi tersebut. 

Pembahasan Soal:

Dengan menggunakan formula binomial newton 

begin mathsize 14px style open parentheses straight a plus straight b close parentheses to the power of straight n equals sum from straight r equals 0 to straight n of straight C subscript straight r superscript straight n times straight a to the power of straight n minus straight r end exponent times straight b to the power of straight r end style 

Maka 

begin mathsize 14px style open parentheses 1 plus 1 fourth close parentheses to the power of straight n equals sum from straight r equals 0 to straight n of straight C subscript straight r superscript straight n times open parentheses 1 close parentheses to the power of straight n minus straight r end exponent times open parentheses 1 fourth close parentheses to the power of straight r end style 

Untuk suku ketiga begin mathsize 14px style open parentheses straight r equals 2 close parentheses end style dan untuk suku keempat begin mathsize 14px style open parentheses straight r equals 3 close parentheses end style. Suku ketiga adalah dua kali suku keempat maka; 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight C subscript 2 superscript straight n times open parentheses 1 close parentheses to the power of straight n minus 2 end exponent times open parentheses 1 fourth close parentheses squared end cell equals cell 2 times straight C subscript 3 superscript straight n times open parentheses 1 close parentheses to the power of straight n minus 3 end exponent times open parentheses 1 fourth close parentheses cubed end cell row cell fraction numerator straight n factorial over denominator open parentheses straight n minus 2 close parentheses factorial times 2 factorial end fraction times 1 times 1 over 16 end cell equals cell 2 times fraction numerator straight n factorial over denominator open parentheses straight n minus 3 close parentheses factorial times 3 factorial end fraction times 1 times 1 over 64 end cell row cell fraction numerator up diagonal strike straight n factorial end strike over denominator open parentheses straight n minus 2 close parentheses down diagonal strike open parentheses straight n minus 3 close parentheses factorial end strike times 2 factorial end fraction times 1 over 16 end cell equals cell fraction numerator up diagonal strike straight n factorial end strike over denominator down diagonal strike open parentheses straight n minus 3 close parentheses factorial end strike times 3 factorial end fraction times 1 over 32 end cell row cell fraction numerator 1 over denominator open parentheses straight n minus 2 close parentheses times 2 end fraction.1 over 16 end cell equals cell 1 over 6 times 1 over 32 end cell row cell fraction numerator 1 over denominator open parentheses straight n minus 2 close parentheses end fraction times 1 over 32 end cell equals cell 1 over 6 times 1 over 32 end cell row cell fraction numerator 1 over denominator open parentheses straight n minus 2 close parentheses end fraction end cell equals cell 1 over 6 end cell row cell straight n minus 2 end cell equals 6 row straight n equals 8 end table end style 

Jadi nilai begin mathsize 14px style straight n end style adalah undefined.


Untuk begin mathsize 14px style straight n equals 8 end style, suku tengah pada eskpansi adalah suku pada begin mathsize 14px style straight r equals 4 end style, yaitu 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell suku space tengah end cell equals cell straight C subscript 4 superscript 8 times open parentheses 1 close parentheses to the power of 8 minus 4 end exponent times open parentheses 1 fourth close parentheses to the power of 4 end cell row blank equals cell fraction numerator 8 factorial over denominator open parentheses 8 minus 4 close parentheses factorial times 4 factorial end fraction times 1 times 1 over 256 end cell row blank equals cell fraction numerator 8 times 7 times up diagonal strike 6 times 5 times down diagonal strike 4 factorial end strike over denominator down diagonal strike 4 factorial end strike times 4 times up diagonal strike 3 times 2 times 1 end strike end fraction times 1 over 256 end cell row blank equals cell fraction numerator 8 times 7 times 5 over denominator 4 times 256 end fraction end cell row blank equals cell fraction numerator 280 over denominator 1.024 end fraction end cell row blank equals cell 35 over 128 end cell end table end style 

Jadi suku tengah dari ekspansi tersebut adalah begin mathsize 14px style 35 over 128 end style

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 15 Maret 2021

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