Question

Buktikan dengan Induksi Matematika !

Answer 1

Jawab:

induksi

Penjelasan dengan langkah-langkah:

a(n) =  (n+4)/ n(n+1)(n+2)

a(k) = (k+4)/ k(k+1)(k+2)

a(k+1) = (k+1 + 4 ) /  (k+1) (k+1+ 1) (k+1 + 2)

a(k+1) = (k+5) / (k+1)(k+2)(k+3)

,...

p(n) = n(3n +7) / 2 (n+1)(n+2)

p(k) = k(3k +7) / 2 (k+1)(k+2)

p(k+1) = (k+1) {3(k+1) + 7 } / 2 (k+1 +1)(k+1 +2)

p(k+1) = (k+1) (3k+ 10) /  2 (k+2)(k+3)

.....

induksi matematika

p(k+1) = p(k) + a(k+1)

(k+1) (3k+ 10) /  2 (k+2)(k+3) = k(3k +7) / 2 (k+1)(k+2) + (k+5) / (k+1)(k+2)(k+3)

(k+1) (3k+ 10) /  2 (k+2)(k+3) = [k(3k+7)(k+3)  + (k+5)(2)] / 2(k+1)(k+2)(k+3)

(k+1) (3k+ 10) /  2 (k+2)(k+3) = [k(3k²+13k+21)+ 2k+10 ] / 2(k+1)(k+2)(k+3)

(k+1) (3k+ 10) /  2 (k+2)(k+3) = [3k³+ 13k²+ 23k + 10]/ 2(k+1)(k+2)(k+3)

(k+1) (3k+ 10) /  2 (k+2)(k+3) = (k+1)(k+1)((3k+10) / 2(k+1)(k+2)(k+3)

(k+1) (3k+ 10) /  2 (k+2)(k+3) = (k+1)(3k+10) / 2(k+2)(k+3)