Demostración
Demostración
Determinar si
n
(3r-2)(6r+2) = 2n(3n^2+3n-2) n
r=1 18r^2-6r-4 = 6n^3+6n^2-4n n
Trabajaremos con:
(3r-2)(6r+2) = 18r^2-6r-4
2n(3n^2+3n-2) =6n^3+6n^2-4n
Trabajo previo. Para:
P(1) 18(1)^2-6(1)-4 = 6(1)^3+6(1)^2-4(1)
8=8 es "V"
P(2) [18(1)^2-6(1)-4] + [18(2)^2-6(2)-4] = 6(2)^3+6(2)^2-4(2)
8+56=64
64=64 es "V"
P(3) [18(1)^2-6(1)-4] + [18(2)^2-6(2)-4] + [18(3)^2-6(3)-4] = 6(3)^3+6(3)^2-4(3)
8+56+140=204
204=204 es "V"
Determinar para p(k)
[18(1)^2-6(1)-4] + [18(2)^2-6(2)-4] + ... + [18(k)^2-6(k)-4] = 6(k)^3+6(k)^2-4(k)
Entonces p(k+1) 6n^3+6n^2-4n
6(k+1)^3+6(k+1)^2-4(k+1)
Efectuamos productos
6(k^3+3k^2+3k+1)+6(k^2+2k+1)-4(k+1)
6k^3+18k^2+18k+6+6k^2+12k+6-4k-4
6k^3+24k^2+26k+8
Entonces p(k+1)
[18(1)^2-6(1)-4] + [18(2)^2-6(2)-4] + ... + [18(k)^2-6(k)-4] + [18(k+1)^2-6(k+1)-4] = 6(k)^3+6(k)^2-4(k) + [18(k+1)^2-6(k+1)-4]
= 6k^3+6k^2-4k + [18(k+1)^2-6(k+1)-4]
= 6k^3+6k^2-4k + [18k^2+36k+18-6(k+1)-4]
= 6k^3+6k^2-4k + [18k^2+36k+18-6k-6-4]
= 6k^3+6k^2-4k + 18k^2+30k+8
= 6k^3+24k^2+26k+8