Tentukan nilai dari n=1∑∞​2⋅32−n.

Pertanyaan

Tentukan nilai dari sum from n equals 1 to infinity of 2 times 3 to the power of 2 minus n end exponent

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Jawaban

 nilai dari sum from n equals 1 to infinity of 2 times 3 to the power of 2 minus n end exponent equals 9

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah 9.

Terlebih dahulu kita substitusi nilai n equals 1 comma 2 comma 3 comma 4 comma space horizontal ellipsis , maka diperoleh :

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell sum from n equals 1 to infinity of 2 times 3 to the power of 2 minus n end exponent end cell equals cell open parentheses 2 times 3 to the power of 2 minus 1 end exponent close parentheses plus open parentheses 2 times 3 to the power of 2 minus 2 end exponent close parentheses plus open parentheses 2 times 3 to the power of 2 minus 3 end exponent close parentheses plus open parentheses 2 times 3 to the power of 2 minus 4 end exponent close parentheses plus horizontal ellipsis end cell row blank equals cell open parentheses 2 times 3 to the power of 1 close parentheses plus open parentheses 2 times 3 to the power of 0 close parentheses plus open parentheses 2 times 3 to the power of negative 1 end exponent close parentheses plus open parentheses 2 times 3 to the power of negative 2 end exponent close parentheses plus horizontal ellipsis end cell row blank equals cell open parentheses 2 times 3 close parentheses plus open parentheses 2 times 1 close parentheses plus open parentheses 2 times 1 third close parentheses plus open parentheses 2 times 1 over 3 squared close parentheses plus horizontal ellipsis end cell row blank equals cell 6 plus 2 plus 2 over 3 plus 2 over 9 plus horizontal ellipsis end cell end table end style 

Diperoleh deret geometri tak hingga konvergen dengan a equals 6 dan rasio open parentheses r close parentheses yaitu :

table attributes columnalign right center left columnspacing 0px end attributes row r equals cell U subscript 2 over U subscript 1 end cell row blank equals cell 2 over 6 end cell row blank equals cell 1 third end cell end table 

Untuk penyelesaiannya kita gunakan rumus berikut :

table attributes columnalign right center left columnspacing 0px end attributes row cell S subscript infinity end cell equals cell fraction numerator a over denominator 1 minus r end fraction end cell row blank equals cell fraction numerator 6 over denominator 1 minus begin display style 1 third end style end fraction end cell row blank equals cell fraction numerator 6 over denominator begin display style fraction numerator 3 minus 1 over denominator 3 end fraction end style end fraction end cell row blank equals cell fraction numerator 6 over denominator begin display style 2 over 3 end style end fraction end cell row blank equals cell scriptbase up diagonal strike 6 end scriptbase presuperscript 3 cross times fraction numerator 3 over denominator up diagonal strike 2 end fraction end cell row blank equals 9 end table 

Dengan demikian, nilai dari sum from n equals 1 to infinity of 2 times 3 to the power of 2 minus n end exponent equals 9

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