è¿ç±»ä¼¼äºææ³¢é£å¥æ°åãããææ³¢é£å¥æ°åé项å
¬å¼çæ¨å¯¼ããææ³¢é£å¥æ°åï¼1ã1ã2ã3ã5ã8ã13ã21ãâ¦â¦
ããå¦æ设F(n)为该æ°åç第n项(nâN+)ãé£ä¹è¿å¥è¯å¯ä»¥åæå¦ä¸å½¢å¼ï¼
ããF(0) = 0ï¼F(1)=F(2)=1ï¼F(n)=F(n-1)+F(n-2) (nâ¥3)
ããæ¾ç¶è¿æ¯ä¸ä¸ªçº¿æ§éæ¨æ°åã
ããé项å
¬å¼çæ¨å¯¼æ¹æ³ä¸ï¼å©ç¨ç¹å¾æ¹ç¨
ãã线æ§éæ¨æ°åçç¹å¾æ¹ç¨ä¸ºï¼
ããX^2=X+1
ãã解å¾
ããX1=(1+â5)/2,ï¼X2=(1-â5)/2
ããåF(n)=C1*X1^n + C2*X2^n
ããâµF(1)=F(2)=1
ããâ´C1*X1 + C2*X2
ããC1*X1^2 + C2*X2^2
ãã解å¾C1=1/â5ï¼C2=-1/â5
ããâ´F(n)=(1/â5)*{[(1+â5)/2]^n - [(1-â5)/2]^n}ï¼â5è¡¨ç¤ºæ ¹å·5ï¼
ããé项å
¬å¼çæ¨å¯¼æ¹æ³äºï¼æ®éæ¹æ³
ãã设常æ°rï¼s
ãã使å¾F(n)-r*F(n-1)=s*[F(n-1)-r*F(n-2)]
ããår+s=1ï¼ -rs=1
ããnâ¥3æ¶ï¼æ
ããF(n)-r*F(n-1)=s*[F(n-1)-r*F(n-2)]
ããF(n-1)-r*F(n-2)=s*[F(n-2)-r*F(n-3)]
ããF(n-2)-r*F(n-3)=s*[F(n-3)-r*F(n-4)]
ããâ¦â¦
ããF(3)-r*F(2)=s*[F(2)-r*F(1)]
ããå°ä»¥ä¸n-2个å¼åç¸ä¹ï¼å¾ï¼
ããF(n)-r*F(n-1)=[s^(n-2)]*[F(2)-r*F(1)]
ããâµs=1-rï¼F(1)=F(2)=1
ããä¸å¼å¯åç®å¾ï¼
ããF(n)=s^(n-1)+r*F(n-1)
ããé£ä¹ï¼
ããF(n)=s^(n-1)+r*F(n-1)
ãã= s^(n-1) + r*s^(n-2) + r^2*F(n-2)
ãã= s^(n-1) + r*s^(n-2) + r^2*s^(n-3) + r^3*F(n-3)
ããâ¦â¦
ãã= s^(n-1) + r*s^(n-2) + r^2*s^(n-3) +â¦â¦+ r^(n-2)*s + r^(n-1)*F(1)
ãã= s^(n-1) + r*s^(n-2) + r^2*s^(n-3) +â¦â¦+ r^(n-2)*s + r^(n-1)
ããï¼è¿æ¯ä¸ä¸ªä»¥s^(n-1)为é¦é¡¹ã以r^(n-1)为æ«é¡¹ãr/s为å
¬æ¯ççæ¯æ°åçå项çåï¼
ãã=[s^(n-1)-r^(n-1)*r/s]/(1-r/s)
ãã=(s^n - r^n)/(s-r)
ããr+s=1ï¼ -rs=1çä¸è§£ä¸º s=(1+â5)/2ï¼r=(1-â5)/2
ããåF(n)=(1/â5)*{[(1+â5)/2]^n - [(1-â5)/2]^n}
ããè¿ä»£æ³
ããå·²ç¥a1=1,a2=1,an=a(n-1)+a(n-2)(n>=3),æ±æ°å{an}çé项å
¬å¼
ãã解 :设an-αa(n-1)=β(a(n-1)-αa(n-2))
ããå¾Î±+β=1
ããαβ=-1
ããæé æ¹ç¨x²-x-1=0,解å¾Î±=(1-â5)/2,β=(1+â5)/2æα=(1+â5)/2,β=(1-â5)/2
ããæ以
ããan-(1-â5)/2*a(n-1)=(1+â5)/2*(a(n-1)-(1-â5)/2*a(n-2))=[(1+â5)/2]^(n-2)*(a2-(1-â5)/2*a1)`````````1
ããan-(1+â5)/2*a(n-1)=(1-â5)/2*(a(n-1)-(1+â5)/2*a(n-2))=[(1-â5)/2]^(n-2)*(a2-(1+â5)/2*a1)`````````2
ããç±å¼1,å¼2,å¯å¾
ããan=[(1+â5)/2]^(n-2)*(a2-(1-â5)/2*a1)``````````````3
ããan=[(1-â5)/2]^(n-2)*(a2-(1+â5)/2*a1)``````````````4
ããå°å¼3*(1+â5)/2-å¼4*(1-â5)/2,åç®å¾an=(1/â5)*{[(1+â5)/2]^n - [(1-â5)/2]^n}
ãã`````
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