Roboguru

Buktikanlah. c. 1⋅20+2⋅21+3⋅22+4⋅23+....+n⋅2n−1=(n−1)⋅2n+1

Pertanyaan

Buktikanlah.

c. 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight n times 2 to the power of straight n minus 1 end exponent equals left parenthesis straight n minus 1 right parenthesis times 2 to the power of straight n plus 1 

Pembahasan Soal:

Membuktikan dengan menggunakan induksi matematika dimana

Untuk n =1

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight n times 2 to the power of straight n minus 1 end exponent end cell equals cell left parenthesis straight n minus 1 right parenthesis times 2 to the power of straight n plus 1 end cell row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus 1 times 2 to the power of 1 minus 1 end exponent end cell equals cell left parenthesis 1 minus 1 right parenthesis times 2 to the power of 1 plus 1 end cell row cell 1.2 to the power of 0 end cell equals cell 0.2 plus 1 end cell row 1 equals cell 1 rightwards arrow terbukti end cell end table

Untuk n =k diasumsikan terbukti 

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight n times 2 to the power of straight n minus 1 end exponent end cell equals cell left parenthesis straight n minus 1 right parenthesis times 2 to the power of straight n plus 1 end cell row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight k times 2 to the power of straight k minus 1 end exponent end cell equals cell left parenthesis straight k minus 1 right parenthesis times 2 to the power of straight k plus 1 rightwards arrow terbukti end cell end table

Untuk n = k+1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight n times 2 to the power of straight n minus 1 end exponent end cell equals cell left parenthesis straight n minus 1 right parenthesis times 2 to the power of straight n plus 1 end cell row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus open parentheses straight k plus 1 close parentheses times 2 to the power of straight k minus 1 plus 1 end exponent end cell equals cell left parenthesis straight k plus 1 minus 1 right parenthesis times 2 to the power of straight k plus 1 end exponent plus 1 end cell row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus open parentheses straight k plus 1 close parentheses times 2 to the power of straight k end cell equals cell left parenthesis straight k right parenthesis times 2 to the power of straight k plus 1 end exponent plus 1 end cell row cell 2 straight k to the power of straight k plus 2 to the power of straight k end cell equals cell 2 straight k to the power of straight k plus 1 end exponent plus 1 rightwards arrow terbukti end cell end table

Maka akan dibuktikan

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight n times 2 to the power of straight n minus 1 end exponent end cell equals cell left parenthesis straight n minus 1 right parenthesis times 2 to the power of straight n plus 1 end cell row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus k.2 to the power of k plus open parentheses straight k plus 1 close parentheses times 2 to the power of straight k minus 1 plus 1 end exponent end cell equals cell left parenthesis straight k plus 1 minus 1 right parenthesis times 2 to the power of straight k plus 1 end exponent plus 1 end cell row cell 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus 1 left parenthesis straight k minus 1 right parenthesis.2 to the power of straight k plus 2 straight k to the power of straight k plus 2 to the power of straight k end cell equals cell left parenthesis straight k right parenthesis times 2 to the power of straight k plus 1 end exponent plus 1 end cell row cell 1 plus 2 straight k to the power of straight k minus 2 to the power of straight k plus 2 straight k to the power of straight k plus 2 to the power of straight k end cell equals cell 2 straight k to the power of straight k plus 1 end exponent plus 1 end cell row cell 1 plus 4 straight k to the power of straight k end cell equals cell 2 straight k to the power of straight k plus 1 end exponent plus 1 end cell row cell 1 plus 2 straight k.2 to the power of straight k end cell equals cell 2 straight k to the power of straight k plus 1 end exponent plus 1 end cell row cell 1 plus straight k.2 to the power of 1.2 to the power of straight k end cell equals cell 2 straight k to the power of straight k plus 1 end exponent plus 1 end cell row cell 2 straight k to the power of straight k plus 1 end exponent plus 1 end cell equals cell 2 straight k to the power of straight k plus 1 end exponent plus 1 rightwards arrow terbukti end cell end table

Jadi terbukti 1 times 2 to the power of 0 plus 2 times 2 to the power of 1 plus 3 times 2 squared plus 4 times 2 cubed plus.... plus straight n times 2 to the power of straight n minus 1 end exponent equals left parenthesis straight n minus 1 right parenthesis times 2 to the power of straight n plus 1 karena hasil sisi kanan dan kiri sama

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Mahasiswa/Alumni UIN Walisongo Semarang

Terakhir diupdate 06 Oktober 2021

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