Roboguru

Buktikanlah. c.

Pertanyaan

Buktikanlah.

c. sum from straight k equals 1 to straight n of 3 to the power of straight k equals 3 over 2 open parentheses 3 to the power of straight n minus 1 close parentheses 

Pembahasan Soal:

Pembuktian dengan menggunakan induksi matematika dimana untuk n = 1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight k equals 1 to 1 of 3 to the power of 1 end cell equals cell 3 over 2 open parentheses 3 to the power of 1 minus 1 close parentheses end cell row 3 equals cell 3 rightwards arrow terbukti end cell end table

Untuk n = k akan diasumsikan terbukti

table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight k equals 1 to straight k of 3 to the power of straight k end cell equals cell 3 over 2 open parentheses 3 to the power of straight k minus 1 close parentheses rightwards arrow terbukti end cell end table

Untuk n = k+1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight k equals 1 to straight k plus 1 of 3 to the power of straight k end cell equals cell 3 over 2 open parentheses 3 to the power of straight k minus 1 close parentheses plus 3 to the power of left parenthesis straight k plus 1 right parenthesis end exponent end cell row blank equals cell 3 over 2 open parentheses 3 to the power of straight k minus 1 close parentheses plus 3 to the power of straight k.3 to the power of 1 end cell row blank equals cell 3 over 2 open parentheses 3 to the power of straight k minus 1 close parentheses plus 3 open parentheses 3 to the power of straight k close parentheses end cell row blank equals cell 3 over 2 open parentheses 3 to the power of straight k minus 1 close parentheses plus 6 over 2 open parentheses 3 to the power of straight k close parentheses end cell row blank equals cell 3 over 2 open parentheses 3 to the power of straight k close parentheses plus 6 over 2 open parentheses 3 to the power of straight k close parentheses minus 3 over 2 end cell row blank equals cell 9 over 2 open parentheses 3 to the power of straight k close parentheses minus 3 over 2 end cell row blank equals cell fraction numerator 3 squared open parentheses 3 to the power of straight k close parentheses over denominator 2 end fraction minus 3 over 2 end cell row blank equals cell fraction numerator 3 open parentheses 3 to the power of straight k plus 1 end exponent close parentheses over denominator 2 end fraction minus 3 over 2 end cell row blank equals cell 3 over 2 open parentheses 3 to the power of straight k plus 1 end exponent minus 1 close parentheses rightwards arrow terbukti end cell end table

Jadi terbukti bahwa sum from straight k equals 1 to straight n of 3 to the power of straight k equals 3 over 2 open parentheses 3 to the power of straight n minus 1 close parentheses karena hasil sisi kanan dan kiri sama

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 18 Juli 2021

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