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Misalkan diketahui barisan bilangan a1​, a2​, a3​, ..., dengan a1​=2, a2​=5, a3​=8, dan an​=an−1​+an−2​+an−3​. Buktikan bahwa an​≤2n.

Pertanyaan

Misalkan diketahui barisan bilangan begin mathsize 14px style a subscript 1 end stylebegin mathsize 14px style a subscript 2 end stylebegin mathsize 14px style a subscript 3 end style, ..., dengan begin mathsize 14px style a subscript 1 equals 2 end stylebegin mathsize 14px style a subscript 2 equals 5 end stylebegin mathsize 14px style a subscript 3 equals 8 end style, dan begin mathsize 14px style a subscript n equals a subscript n minus 1 end subscript plus a subscript n minus 2 end subscript plus a subscript n minus 3 end subscript end style. Buktikan bahwa begin mathsize 14px style a subscript n less or equal than 2 to the power of n end style

I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Pembahasan

Langkah 1.

Akan dibuktikan begin mathsize 14px style P left parenthesis n right parenthesis end style benar untuk begin mathsize 14px style n equals 1 end style  

begin mathsize 14px style a subscript 1 equals 2 less or equal than 2 to the power of 1 equals 2 end style  

Oleh karena bernilai benar, maka begin mathsize 14px style P left parenthesis n right parenthesis end style benar untuk begin mathsize 14px style n equals 1 end style.

Langkah 2. 

Andaikan benar untuk begin mathsize 14px style n equals 1 , 2 comma 3 comma times times times comma k – 1 comma k comma end style yaitu

a subscript 1 less or equal than 2 to the power of 1 comma a subscript 2 less or equal than 2 squared comma a subscript 3 less or equal than 2 cubed comma times times times comma a subscript k – 2 end subscript less or equal than 2 to the power of k – 2 end exponent comma a subscript k – 1 end subscript less or equal than 2 to the power of k – 1 end exponent.

Akan dibuktikan begin mathsize 14px style P left parenthesis n right parenthesis end style benar untuk begin mathsize 14px style n equals k plus 1 end style, yaitu begin mathsize 14px style a subscript k plus 1 end subscript less or equal than 2 to the power of k plus 1 end exponent end style.

Bukti:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a subscript k plus 1 end subscript end cell equals cell a subscript left parenthesis k plus 1 right parenthesis end subscript plus a subscript left parenthesis k plus 1 right parenthesis – 2 end subscript plus a subscript left parenthesis k plus 1 right parenthesis – 3 end subscript end cell row blank equals cell a subscript k plus a subscript k – 1 end subscript plus a subscript k – 2 end subscript less or equal than 2 to the power of k end cell row blank blank cell plus 2 to the power of k – 1 end exponent plus 2 to the power of k – 2 end exponent end cell row blank equals cell 2 to the power of k plus 2 to the power of k cross times 2 to the power of – 1 end exponent plus 2 to the power of k cross times 2 to the power of – 2 end exponent end cell row blank equals cell 2 to the power of k plus 1 half cross times 2 to the power of k plus 1 fourth cross times 2 to the power of k end cell row blank equals cell 1 3 over 4 cross times 2 to the power of k end cell row blank less or equal than cell 2 cross times 2 to the power of k end cell row blank equals cell 2 to the power of k plus 1 end exponent end cell end table end style  

Dengan demikian terbukti .

Oleh karena Langkah 1 dan Langkah 2 keduanya bernilai benar, maka terbukti bahwabegin mathsize 14px style a subscript n less or equal than 2 to the power of n end style.

184

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