P(n):1*4+ 2*7+ 3*10+...+n(3n+1)=n(n+1)²,∀n∈N*
| Verificarea: P(1):1*4=1(1+1)²<=>4=2²<=>4=4 (A)
P(2):1*4+2*7=2(2+1)²<=>4+14=2*3²<=>18=9*2<=>18=18 (A)
|| Demonstratia:P(k)->P(k+1),∀k∈N*
fie v(P(k))=1
P(k):1*4+ 2*7+ 3*10+...+k(3k+1)=k(k+1)²,∀k∈N*
P(k+1):1*4+ 2*7+ 3*10+...+(n+1)(3n+3+1)=(n+1)(n+1+1)²,∀k∈N*<=>P(k+1):1*4+ 2*7+ 3*10+...+(k+1)(3k+4)=(k+1)(k+2)²,∀k∈N*
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P(k+1):1*4+ 2*7+ 3*10+...+k(3k+1)+(k+1)(3k+4)=k(k+1)²+(k+1)(3k+4)=(k+1)[3k+4+k(k+1)]=(k+1)(3k+4+k²+k)=(k+1)(k²+4k+4)=(k+1)(k+2)² ,∀k∈N*=>v(P(k+1))=1,v(P(k))=1=>P(k)->P(k+1)=>v(P(n))=1
