Pertanyaan

Buktikanlah. a. i = 1 ∑ n a n − 1 ⋅ b n − 1 = a − b a n − b n

Buktikanlah.

a. $i = 1 \sum n a^{n - 1} \cdot b^{n - 1} = \frac{a ^{n} - b ^{n}}{a - b}$

A. Acfreelance

Master Teacher

Mahasiswa/Alumni UIN Walisongo Semarang

Jawaban terverifikasi

Jawaban

terbukti bahwa karena hasil sisi kira dan kanan sama

terbukti bahwa $sum from straight i equals 1 to straight n of straight a to the power of straight n minus 1 end exponent times straight b to the power of straight n minus 1 end exponent equals fraction numerator straight a to the power of straight n minus straight b to the power of straight n over denominator straight a minus straight b end fraction$ karena hasil sisi kira dan kanan sama

Pembahasan

Untuk n = 1 maka Untuk n = k diasumsikan benar maka Akan dibuktikan untuk n = k+1 maka Jadi terbukti bahwa karena hasil sisi kira dan kanan sama

Untuk n = 1 maka

Untuk n = k diasumsikan benar maka

Akan dibuktikan untuk n = k+1 maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell sum from straight i equals 1 to straight n of straight a to the power of straight n minus 1 end exponent times straight b to the power of straight n minus 1 end exponent end cell equals cell fraction numerator straight a to the power of straight n minus straight b to the power of straight n over denominator straight a minus straight b end fraction end cell row cell sum from straight i equals 1 to k plus 1 of straight a to the power of k plus 1 minus 1 end exponent times straight b to the power of k plus 1 minus 1 end exponent end cell equals cell sum from straight i equals 1 to k of straight a to the power of k minus 1 end exponent times straight b to the power of k minus 1 end exponent plus sum from straight i equals straight k plus 1 to straight k plus 1 of straight a to the power of straight k plus 1 minus 1 end exponent times straight b to the power of straight k plus 1 minus 1 end exponent end cell row cell fraction numerator straight a to the power of straight k plus 1 end exponent minus straight b to the power of straight k plus 1 end exponent over denominator straight a minus straight b end fraction end cell equals cell fraction numerator straight a to the power of straight k minus straight b to the power of straight k over denominator straight a minus straight b end fraction plus straight a to the power of straight k. straight b to the power of straight k end cell row cell fraction numerator straight a to the power of straight k plus 1 end exponent minus straight b to the power of straight k plus 1 end exponent over denominator straight a minus straight b end fraction end cell equals cell fraction numerator straight a to the power of straight k minus straight b to the power of straight k over denominator straight a minus straight b end fraction plus fraction numerator open parentheses straight a minus straight b close parentheses straight a to the power of straight k. straight b to the power of straight k over denominator straight a minus straight b end fraction end cell row cell fraction numerator straight a to the power of straight k plus 1 end exponent minus straight b to the power of straight k plus 1 end exponent over denominator straight a minus straight b end fraction end cell equals cell fraction numerator straight a to the power of straight k minus straight b to the power of straight k over denominator straight a minus straight b end fraction plus fraction numerator open parentheses straight a minus straight b close parentheses straight a to the power of straight k. straight b to the power of straight k over denominator straight a minus straight b end fraction end cell row cell fraction numerator straight a to the power of straight k plus 1 end exponent minus straight b to the power of straight k plus 1 end exponent over denominator straight a minus straight b end fraction end cell equals cell fraction numerator straight a to the power of straight k plus 1 end exponent minus straight b to the power of straight k plus 1 end exponent over denominator straight a minus straight b end fraction rightwards arrow benar end cell row blank blank blank end table$

Jadi terbukti bahwa $sum from straight i equals 1 to straight n of straight a to the power of straight n minus 1 end exponent times straight b to the power of straight n minus 1 end exponent equals fraction numerator straight a to the power of straight n minus straight b to the power of straight n over denominator straight a minus straight b end fraction$ karena hasil sisi kira dan kanan sama

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