1å©ç¨å¾
å®å¸¸æ°æ³(éç¹ï¼
ä¾1 å·²ç¥æ°åï½n ï½ä¸ï¼è¥1=1ï¼ä¸n+1=3n-4(n=1,2,3,â¦). æ±æ°åçé项å
¬å¼n.
åæï¼è¥å
³ç³»å¼æ¯n+1=3nå³ä¸ºçæ¯æ°åï¼å æ¤èèå¤ç-4ï¼è¥è½å为n+1+x=3(n+x),åå¯æé çæ¯æ°åï½n+xï½ã
解ï¼è®¾n+1=3n-4æçå形为n+1+x=3ï¼n+x)ï¼å³n+1=3n+2x,æ¯è¾ç³»æ°å¾ï¼x=-2
n+1-2=3(n-2)
æ°åï½n-2ï½æ¯ä»¥1-2=-1为é¦é¡¹ï¼å
¬æ¯ä¸º3ççæ¯æ°å
n-2=ï¼-1ï¼3n-1 å³n = -3n-1+2.
说æï¼ç»åºä¸é¶éæ¨å
³ç³»å¼å½¢å¦ (n=1,2,â¦),AãB为常æ°ï¼åå¯ä½¿ç¨å¾
å®å¸¸æ°æ³ï¼æé çæ¯æ°åæ±åºé项ã
ä¾2 å·²ç¥æ°åï½n ï½ä¸ï¼ån项åsn = 2n-3n, æ±æ°åçé项å
¬å¼n.
åæï¼å·²ç¥çå¼ä¸ä¸æ¯éæ¨å
³ç³»å¼ï¼å©ç¨å¯è½¬å为ï¼n -2n-1=2,èè3n-1æ¯åéï¼å¼å
¥å¾
å®å¸¸æ°xæ¶ï¼å¯è®¾n- x=2(n-1- x),ä»èå¯æé çæ¯æ°åã
解ï¼1=s1=21-3 å1=3ï¼
å½n2æ¶ï¼ =ï¼2n-3nï¼-ï¼2n-1-3n-1ï¼å³n-2n-1=2 ,设å
¶å¯æçå形为ï¼n- x=2(n-1- x)ï¼ï¼éè¦æ³¨æçæ¯ä¸é¢çææ°ï¼è¿æ¯æç§å
³ç³»èä¸æ¯åºå®ç常æ°ï¼æ
å¨æçåå½¢æ¶é注æ两边对åºçå
³ç³»ï¼èä¸åºè¯¥ç¨X代æ¿xï¼ä¹å¯ä»¥ä¸è®¾â-â设â+âï¼ç»ææ¯ä¸æ ·çãï¼
å³ n -2n-1=x ,æ¯è¾ç³»æ°å¾ï¼x=2.
n- 2=2(n-1- 2 )
æ°åï½n- 2ï½æ¯ä»¥1-6=-3为é¦é¡¹ï¼å
¬æ¯ä¸º2ççæ¯æ°åã
n- 2=ï¼-3ï¼2n-1
n=2-3.
说æï¼å¯¹äºåå¦n=An-1+f(n)ï¼A为常æ°ï¼çä¸é¶éæ¨å
³ç³»å¼ãå¯å©ç¨å¾
å®å¸¸æ°æ³ï¼æé çæ¯æ°åï¼ä½é¡»ä½ç°æ°æ°åç¸é»ä¸¤é¡¹çè§å¾æ§ï¼è®¾å
¶å¯æçå形为ï¼n- xg(n)=A[n-1- xg(n-1)],è¥xåå¨,åå¯æé çæ¯æ°å{ n- xg(n)}ã
2 å©ç¨é
æ¹æ³
æäºéæ¨å
³ç³»å¼ç»âé
æ¹âåï¼å¯ä½ç°çå·®ï¼æ¯ï¼çè§å¾æ§ã
ä¾3 设n0ï¼1=5ï¼å½n2æ¶ï¼n+n-1=+6, æ±æ°åçé项å
¬å¼nã
åæï¼ç»åºçéæ¨å
³ç³»å¼ä¸è½åæ è§å¾æ§ï¼å æ¤èèå»åæ¯å¾ï¼2n-2n-1=7+6(n-n-1),为ä½ç°è§å¾æ§ï¼å形为ï¼2n-2n-1-6n+6n-1=7ï¼å³ï¼n-3ï¼2-(n-1-3)2=7.
解ï¼ç±n+n-1=+6ï¼n2ï¼å形为ï¼
2n-2n-1=7+6(n-n-1) å³ï¼n-3ï¼2-(n-1-3)2=7 ï¼n2ï¼
æ°å{ }æ¯ä»¥(1-3)2=4为é¦é¡¹ï¼å
¬å·®ä¸º7ççå·®æ°å
=4+7ï¼n-1ï¼=7n-3ï¼èn0
n=+3
说æï¼éæ¨å
³ç³»å¼ä¸å«æäºæ¬¡é¡¹ãä¸æ¬¡é¡¹æ¶å¯èèç¨é
æ¹æ³ï¼æ示è§å¾ï¼æé çå·®ï¼æ¯ï¼æ°åã
3 å©ç¨å å¼å解
æäºéæ¨å
³ç³»å¼ç»å å¼å解åï¼å¯ä½ç°çå·®ï¼æ¯ï¼çè§å¾æ§ã
ä¾4å·²ç¥æ°åï½n ï½æ¯é¦é¡¹ä¸º1çæ£é¡¹æ°åï¼ä¸2n+1 + 3n+1 - 22n + 3n - nn+1=0æ±æ°åçé项å
¬å¼nã
åæ:ç±å·²ç¥éæ¨å
³ç³»å¼ï¼è¥é
æ¹ï¼åæ æ³é
æå®å
¨å¹³æ¹æå®å
¨å¹³æ¹é¡¹ä¹åãå æ¤èèç¨å å¼å解åç®ï¼å¯»æ±æ´å®è´¨çå
³ç³»ãå¯å形为ï¼n+1ï¼n+1 +3ï¼+3n - nn+1 +nï¼-2nï¼=0ã
解ï¼ç±å·²ç¥æï¼n+1ï¼n+1 +3ï¼+3n - nn+1 +nï¼-2 nï¼=0
ï¼n+1 + nï¼[ï¼n+1 + 3ï¼-2n]=0ï¼èn0
n+1 + 3 -2n=0ï¼åå©ç¨å¾
å®å¸¸æ°æ³æï¼n+1 - 3ï¼-2ï¼n -3ï¼=0
æ°å{n -3}æ¯ä»¥1-3=-2为é¦é¡¹ï¼å
¬æ¯ä¸º2ççæ¯æ°åã
n-3 =ï¼-2ï¼2n-1 å³n = 3-2n
说æï¼å å¼å解è½è¾¾å°åç®çç®çï¼ä½¿éæ¨å
³ç³»å¼ç®åï¼å¸æ¾è§å¾æ§ã
5 å©ç¨åæ°
æäºæ°åçéæ¨å
³ç³»å¼ï¼ç»ååæ°åå½¢åï¼æ¾ç°åºè§å¾æ§ï¼å¯æé çæ¯ï¼å·®ï¼æ°åã
ä¾7 å·²ç¥x1=1,x2=2,xn+ 2=,è¯æ±xn ã
åæï¼ç±éæ¨å
³ç³»å¼ç»æç¹å¾ï¼æèæ³å°åæ°ï¼å³æ xn+2 =ï¼ä»è
=ï¼å¯æé çæ¯æ°åã
解ï¼å¯¹éæ¨å
³ç³»å¼ä¸¤è¾¹ååæ°å¾ï¼=
å¯å形为=ï¼-ï¼ï¼ï¼
æ°å{}æ¯ä»¥=-为é¦é¡¹ï¼å
¬æ¯ä¸º-ççæ¯æ°å
=ï¼-ï¼(-= (n2)
=+()+()+ ⦠+()
= 1 + ï¼-ï¼+ï¼-ï¼2 + ⦠+
= + (n2)
= (n2) èå½n=1æ¶äº¦æ»¡è¶³ã
= (n1)
说æï¼éæ¨å
³ç³»å¼ä¸å«æç¸é»ä¸¤é¡¹ä¹ç§¯ä¸ç¸é»ä¸¤é¡¹ä¹åçå
³ç³»ï¼å¯èèååæ°ï¼æå为åå¼ï¼ï¼æ示è§å¾ï¼æé çæ¯ï¼å·®ï¼æ°åã
ä¾8å·²ç¥æ°åï½n ï½ä¸ï¼1=7ï¼n2æ¶ï¼ï¼æ±æ°åçé项å
¬å¼n
åæï¼å·²ç¥éæ¨å
³ç³»å¼å³è¾¹ä¸ºåå¼ï¼ååæ°åå¯å为ï¼ï¼æªè½åæ è§å¾ï¼
ä½è¥è½å为çå
³ç³»ï¼åå¯æ示è§å¾ï¼ç»åå¾
å®å¸¸æ°æ³ï¼å¯ç¡®å®Aå¼ã
解ï¼ç±å·²ç¥ï¼ ï¼A0ï¼å³(2A+1â 0)
令ï¼è§£å¾ï¼A=1
å·²ç¥å
³ç³»å¼å¯æçå形为ï¼ååæ°å¾ï¼ ï¼n2ï¼ã
æ°å{}æ¯ä»¥=为é¦é¡¹ï¼å
¬å·®ä¸ºççå·®æ°åã
= +ï¼n-1ï¼,å³ ï¼n1ï¼
说æï¼â ä¾8ä¸çéæ¨å
³ç³»å¼ç»æç¹å¾ï¼äº¦ææ³å°ååæ°ï¼ä½è¦çµæ´»ç»åå¾
å®å¸¸æ°æ³ï¼æé æ°æ°åï¼å¸æ¾çå·®çè§å¾æ§ã
â¡å¼å
¥å¾
å®å¸¸æ°Aæ¯ä¸ºäºæ示ååçä¸è´æ§ï¼è§å¾æ§ï¼ï¼è¥Aå¼åå¨ï¼åå¯åæ æ¤ååè§å¾ãè¥Aå¼ä¸åå¨ï¼åèèå
¶å®åå½¢ã
6 å©ç¨æ¢å
æäºæ°åçéæ¨å
³ç³»å¼çèµ·æ¥è¾ä¸ºå¤æï¼ä½åºç¨æ¢å
ååå½ææ³åï¼å¯æé æ°æ°åè¿è¡ä»£æ¢ï¼ä½¿éæ¨å
³ç³»å¼ç®åï¼ä»èæ示çå·®ï¼æ¯ï¼è§å¾ï¼æ±åºé项ã
ä¾9å·²ç¥æ°åï½an ï½ä¸ï¼ æ±ï¼1981年第22å±IMOé¢éé¢ï¼ã
åæï¼å·²ç¥éæ¨å
³ç³»å¼ä¸çè¾é¾å¤çï¼èèç¨æ¢å
å»ææ ¹å¼ï¼å³ä»¤ï¼0ï¼ã
解ï¼ä»¤ï¼å=5ï¼ 0ï¼ä»è=
ç±å·²ç¥éæ¨å
³ç³»å¼æï¼
åç®å¾ï¼=ï¼ï¼2
2=ï¼ ç±å¾
å®å¸¸æ°æ³å¾ï¼2ï¼-3ï¼= -3
æ°å{-3}æ¯ä»¥-3=2为é¦é¡¹ï¼å
¬æ¯ä¸ºççæ¯æ°åã
-3=2ï¼ï¼n-1 å³ = + 3
== (n1)
说æï¼å¯¹äºéæ¨å
³ç³»å¼ä¸è¾é¾å¤ççæ ¹å¼ï¼æ¯å¦ä¸è½åæ ç¸é»é¡¹çè§å¾æ§ï¼ï¼å¯éç¨æ¢å
å»ææ ¹å¼ï¼åç®éæ¨å
³ç³»å¼ï¼æ示ç¸é»é¡¹çååè§å¾ï¼æé çæ¯ï¼å·®ï¼æ°åã
ä¾10 设=1ï¼=ï¼nNï¼,æ±è¯ï¼ï¼1990å¹´åçå©å¥¥æå¹å
è¯é¢ï¼ã
åæï¼æ¯è¾å·²ç¥ä¸ç»è®ºï¼åºå
æ±é项å
¬å¼ãå¾
è¯çä¸çå¼ä¸å«æï¼ä¸å·²ç¥éæ¨å
³ç³»å¼ä¸å«æï¼æ®æ¤ä¸¤ä¸ªä¿¡æ¯ï¼èèè¿è¡ä¸è§ä»£æ¢ï¼åç®éæ¨å
³ç³»å¼ã
è¯æï¼ç±å·²ç¥0ï¼å¼å
¥æ°å{}使=tan,(0,)
ç±å·²ç¥æï¼=
å³=ï¼å=1ï¼ï¼ä»è
å³æ°å{}æ¯ä»¥ä¸ºé¦é¡¹ï¼å
¬æ¯ä¸ºççæ¯æ°å
= = ï¼ èå½x(0,)æ¶ï¼ætanxX
= tan
温馨提示:答案为网友推荐,仅供参考