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1. å
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çå·®æ°åæ±åå
¬å¼:Sn=n(a1+an)/2=na1+n(n-1)d/2
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¬å¼ï¼Sn=na1(q=1)
Sn=a1(1-qn)/(1-q)=(a1-anÃq)/(1-q) (qâ 1)
2.éä½ç¸åæ³
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{ an }ã{ bn }åå«æ¯çå·®æ°ååçæ¯æ°å. Sn=a1b1+a2b2+a3b3+...+anbn
ä¾å¦ï¼
an=a1+(n-1)d
bn=a1•q(n-1)
Cn=anbn
Tn=a1b1+a2b2+a3b3+a4b4....+anbn
qTn= a1b2+a2b3+a3b4+...+a(n-1)bn+anb(n+1)
Tn-qTn= a1b1+b2(a2-a1)+b3(a3-a2)+...bn[an-a(n-1)]-anb(n+1)
Tn(1-q)=a1b1-anb(n+1)+d(b2+b3+b4+...bn)
=a1b1-an•b1•qn+d•b2[1-q(n-1)]/(1-q)
Tn=ä¸è¿°å¼å/(1-q)
3.ååºç¸å æ³
è¿æ¯æ¨å¯¼çå·®æ°åçån项åå
¬å¼æ¶æç¨çæ¹æ³ï¼å°±æ¯å°ä¸ä¸ªæ°ååè¿æ¥æåï¼ååºï¼ï¼åæå®ä¸åæ°åç¸å ï¼å°±å¯ä»¥å¾å°n个(a1+an)
Sn =a1+ a2+ a3+...... +an
Sn =an+ a(n-1)+a(n-3)...... +a1
ä¸ä¸ç¸å å¾å°2Sn å³ Sn= ï¼a1+an)n/2
4.åç»æ³
æä¸ç±»æ°åï¼æ¢ä¸æ¯çå·®æ°åï¼ä¹ä¸æ¯çæ¯æ°åï¼è¥å°è¿ç±»æ°åéå½æå¼ï¼å¯å为å 个çå·®ãçæ¯æ常è§çæ°åï¼ç¶ååå«æ±åï¼åå°å
¶å并å³å¯.
ä¾å¦ï¼an=2n+n-1
5.è£é¡¹æ³
éç¨äºåå¼å½¢å¼çé项å
¬å¼ï¼æä¸é¡¹ææ两个æå¤ä¸ªçå·®çå½¢å¼ï¼å³an=f(n+1)ï¼f(n)ï¼ç¶åç´¯å æ¶æµæ¶ä¸é´ç许å¤é¡¹ã
常ç¨å
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ï¼1ï¼1/n(n+1)=1/n-1/(n+1)
ï¼2ï¼1/(2n-1)(2n+1)=1/2[1/(2n-1)-1/(2n+1)]
ï¼3ï¼1/n(n+1)(n+2)=1/2[1/n(n+1)-1/(n+1)(n+2)]
ï¼4ï¼1/(âa+âb)=[1/(a-b)](âa-âb)
ï¼5ï¼ n•n!=(n+1)!-n!
[ä¾] æ±æ°åan=1/n(n+1) çån项å.
解ï¼an=1/n(n+1)=1/n-1/(n+1) ï¼è£é¡¹ï¼
å
Sn
=1-1/2+1/2-1/3+1/4â¦+1/n-1/(n+1)ï¼è£é¡¹æ±åï¼
ï¼ 1-1/(n+1)
ï¼ n/(n+1)
å°ç»ï¼æ¤ç±»åå½¢çç¹ç¹æ¯å°åæ°åæ¯ä¸é¡¹æ为两项ä¹åï¼å
¶ä¸ä¸é´ç大é¨å项é½äºç¸æµæ¶äºãåªå©ä¸æéçå 项ã
注æï¼ ä½ä¸ç项å
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2ä½ä¸ç项ååçæ£è´æ§æ¯ç¸åçã
6.æ°å¦å½çº³æ³
ä¸è¬å°ï¼è¯æä¸ä¸ªä¸æ£æ´æ°næå
³çå½é¢ï¼æå¦ä¸æ¥éª¤ï¼
ï¼1ï¼è¯æå½nå第ä¸ä¸ªå¼æ¶å½é¢æç«ï¼
ï¼2ï¼å设å½n=kï¼kâ¥nç第ä¸ä¸ªå¼ï¼k为èªç¶æ°ï¼æ¶å½é¢æç«ï¼è¯æå½n=k+1æ¶å½é¢ä¹æç«ã
ä¾ï¼
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1Ã2Ã3Ã4 + 2Ã3Ã4Ã5 + 3Ã4Ã5Ã6 + â¦â¦ + n(n+1)(n+2)(n+3) = [n(n+1)(n+2)(n+3)(n+4)]/5
è¯æï¼
å½n=1æ¶ï¼æï¼
1Ã2Ã3Ã4 + 2Ã3Ã4Ã5 = 2Ã3Ã4Ã5Ã(1/5 +1) = 2Ã3Ã4Ã5Ã6/5
å设å½é¢å¨n=kæ¶æç«ï¼äºæ¯ï¼
1Ã2Ã3Ã4 + 2Ã3Ã4Ã5 + 3Ã4Ã5Ã6 + â¦â¦ + k(k+1)(k+2)(k+3) = [k(k+1)(k+2)(k+3)(k+4)]/5
åå½n=k+1æ¶æï¼
1Ã2Ã3Ã4 + 2Ã3Ã4Ã5 + 3Ã4Ã5Ã6 + â¦â¦ + (k+1)(k+2)(k+3)(k+4)
= 1Ã2Ã3Ã4 + 2Ã3Ã4*5 + 3Ã4Ã5Ã6 + â¦â¦ + k(k+1)(k+2)(k+3) + (k+1)(k+2)(k+3)(k+4)
= [k(k+1)(k+2)(k+3)(k+4)]/5 + (k+1)(k+2)(k+3)(k+4)
= (k+1)(k+2)(k+3)(k+4)*(k/5 +1)
= [(k+1)(k+2)(k+3)(k+4)(k+5)]/5
å³n=k+1æ¶åçå¼ä»ç¶æç«ï¼å½çº³å¾è¯
7.é项åå½
å
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å¦ï¼æ±æ°å1ï¼1+2ï¼1+2+3ï¼1+2+3+4,â¦â¦çån项åãæ¤æ¶å
å°anæ±åºï¼åå©ç¨åç»çæ¹æ³æ±åã
8.并项æ±åï¼
ä¾ï¼1ï¼2+3ï¼4+5ï¼6+â¦â¦+ï¼2n-1ï¼-2n
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æ¹æ³äºï¼
ï¼1ï¼2ï¼+ï¼3ï¼4ï¼+ï¼5ï¼6ï¼+â¦â¦+[ï¼2n-1ï¼-2n]
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