證明;用數(shù)學(xué)歸納法
1,當(dāng)n=1時(shí)
P(1,1)=1
P(2,2)-1=2*1-1=1
P(1,1)= P(2,2)-1成立
2,假設(shè)n=K,k屬于N成立,
即P1^1+2*P2^2+3*P3^3+...k*Pk^k=P(k+1)^(k+1)-1成立
則當(dāng)n=K+1時(shí)
左邊=P1^1+2*P2^2+3*P3^3+...+k*Pk^k+(k+1)P(k+1)^(k+1)
=P(k+1)^(k+1)-1+(k+1)P(k+1)^(k+1)
=(k+1)!-1+(k+1)*(k+1)!
=(k+1)!(k+2)-1
=(k+2)!-1
=P(k+2)^(k+2)-1
=右邊
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