Tentukan nilai dari n=1∑∞​5⋅2n32​.

Pertanyaan

Tentukan nilai dari sum from n equals 1 to infinity of 5 times 32 over 2 to the power of n.  

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Jawaban

 nilai dari sum from n equals 1 to infinity of 5 times 32 over 2 to the power of n equals 160.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah 160.

Terlebih dahulu kita substitusi nilai n equals 1 comma space 2 comma space 3 comma space 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 5 times 32 over 2 to the power of n end cell equals cell open parentheses 5 times 32 over 2 to the power of 1 close parentheses plus open parentheses 5 times 32 over 2 squared close parentheses plus open parentheses 5 times 32 over 2 cubed close parentheses plus open parentheses 5 times 32 over 2 to the power of 4 close parentheses plus horizontal ellipsis end cell row blank equals cell open parentheses 5 times 32 over 2 close parentheses plus open parentheses 5 times 32 over 4 close parentheses plus open parentheses 5 times 32 over 8 close parentheses plus open parentheses 5 times 32 over 16 close parentheses plus horizontal ellipsis end cell row blank equals cell open parentheses 5 times 16 close parentheses plus open parentheses 5 times 8 close parentheses plus open parentheses 5 times 4 close parentheses plus open parentheses 5 times 2 close parentheses plus horizontal ellipsis end cell row blank equals cell 80 plus 40 plus 20 plus 10 plus horizontal ellipsis end cell end table end style 

Diperoleh deret geometri tak hingga konvergen dengan a equals 80 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 40 over 80 end cell row blank equals cell 1 half 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 80 over denominator 1 minus begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator 80 over denominator begin display style fraction numerator 2 minus 1 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator 80 over denominator begin display style 1 half end style end fraction end cell row blank equals cell 80 cross times 2 over 1 end cell row blank equals 160 end table 

Dengan demikian, nilai dari sum from n equals 1 to infinity of 5 times 32 over 2 to the power of n equals 160.

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