Răspunsuri

2014-09-09T17:31:18+03:00
1) verificam P(1)=1·2=(1/3)·(1+1)(1+2) ⇒ 2=(1/3)·2·3=2 deci adevarata pt P(1)
Presupunem P(n) adevarata si demontram ca P(n+1) adevarata
P(n+1) = 1·2+2·3+....+n(n+1)+(n+1)(n+2)=P(n)+(n+1)(n+2)=(n/3)·(n+1)(n+2)+(n+1)(n+2)=(n+1)(n+2)[(n/3)+1]=(n+1)(n+2)[(n+3)/3]=[(n+1)/3](n+2)(n+3) deci P(n+1) adevarata

2) P(1)=(6²-1)=35 divizibil cu 35
Presupunem P(n) adevarata si demontram ca P(n+1) adevarata
P(n+1) = (6^{2n+2}-1) =  6^{2n}*6^{2}-1=(6^{2n}-1)*6^{2}+6^{2}-1=P(n)*6^{2}+35      unde P(n) este divizibil cu 35 + 35 este un numar divizibil cu 35 deci P(n+1) adevarata 

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