#1
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͹ءÃÁ¤ÃѺ
1. ¡Ó˹´ÅӴѺ $-1 ,\frac{1}{4},\frac{3}{7},\frac{1}{2},...$ â´ÂÁÕ $a_{30}=\frac{57}{88}$ áÅéÇ $a_{24}$ ÁÕ¤èÒà·èÒã´
2. ¶éÒ $a_n$ à»ç¹ÅӴѺ«Öè§ $a_n>0$ áÅÐ $\frac{a_{n+1}^2}{a_{n+1}+2a_n}=a_n$ ÊÓËÃѺ·Ø¡¨Ó¹Ç¹àµçÁºÇ¡ n áÅéÇ $\frac{1}{a_1}\sum_{n = 1}^{10}a_n$ ÁÕ¤èÒà·èҡѺà·èÒã´ 3.ãËé $S_n$ á·¹¼ÅºÇ¡¢Í§Í¹Ø¡ÃÁàÅ¢¤³ÔµªØ´·Õè 1 áÅÐ $Z_n$ á·¹¼ÅºÇ¡¢Í§Í¹Ø¡ÃÁàÅ¢¤³ÔµªØ´·Õè2 áÅжéÒ $\frac{S_n}{Z_n}=\frac{7n+1}{4n+27}$ áÅéÇÍѵÃÒÊèǹÃÐËÇèÒ§¾¨¹ì·Õè 11 ¢Í§ªØ´·Õè 1 ¡ÑºªØ´·Õè2 ÁÕ¤èÒà·èÒã´ *Í.¼Áà©ÅÂãËé´ÙáÅéǤÃѺ áµèÍÂÒ¡ãËé·Ø¡·èÒ¹ä´éÅͧ·ÓÍèÒ¤ÃѺ
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#2
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ÍéÒ§ÍÔ§:
$a_{n+1}^2-a_na_{n+1}-2a_n^2=0$ $(a_{n+1}-2a_n)(a_{n+1}+a_n)=0$ $a_{n+1}=2a_n,-a_n$ áµè$a_n>0$ ÊÓËÃѺ·Ø¡¨Ó¹Ç¹àµçÁºÇ¡ n ´Ñ§¹Ñé¹$a_{n+1}\not= -a_n$ $a_{n+1}=2a_n$ $a_2=2a_1$ $a_3=2a_2=4a_1$ $a_4=2a_3=8a_1$ $a_n=2^{n-1}a_1$ $\frac{1}{a_1}\sum_{n = 1}^{10}a_n$ $=\frac{1}{a_1}\left(\,a_1+a_2+a_3+...+a_{10}\right) $ $=\frac{1}{a_1}\left(\,a_1+2a_1+4a_1+...+2^9a_1\right) $ $=1+2+2^2+2^3+...+2^9$ $=2^{10}-1$
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"¶éÒàÃÒÅéÁºèÍÂæ ã¹·ÕèÊØ´àÃÒ¨ÐÃÙéÇèÒ¶éÒ¨ÐÅéÁ ÅéÁ·èÒä˹¨Ðà¨çº¹éÍ·ÕèÊØ´ áÅÐÃÙéÍÕ¡ÇèÒµèÍä»·ÓÂѧ䧨ÐäÁèãËéÅéÁÍÕ¡ ´Ñ§¹Ñ鹨§ÍÂèÒ¡ÅÑÇ·Õè¨ÐÅéÁ"...ÍÒ¨ÒÃÂìÍӹǠ¢¹Ñ¹ä·Â ¤ÃÑé§áá㹪ÕÇÔµ·ÕèÊͺ¤³ÔµÊÁÒ¤Á¤³ÔµÈÒʵÃìàÁ×èÍ»Õ2533...¼Áä´éá¤è24¤Ðá¹¹(¨Ò¡ÃéͤÐá¹¹) |
#3
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¨Ò¡ Sn/Zn = (7n+1)/(4n+27)
á·¹ n = 21 ¨Ðä´éÇèÒ (2a1+20d)/(2b1+20d') = (7*21+1)/(4*21+27) (a1+10d)/(b1+10d') = (7*21+1)/(4*21+27) a11/b11 = (7*21+1)/(4*21+27) = 148/111 |
#4
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¢éÍáá¹ÕèÍÂÒ¡ãËéµÍºÇèÒ $a_n=\dfrac{2n-3}{3n-2}$ ËÃ×Íà»ÅèÒ¤ÃѺ
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#5
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äÁèãªè¤ÃѺ à¡×ͺ¶Ù¡áÅéǤÃѺ µéͧ¡Òäӵͺà»ç¹µÑÇàÅ¢¤ÃѺ ¾¨¹ì·Õè 24 ÍèÒ¤ÃѺ ⨷Âì¿Ô¡«ì¾¨¹ì·Õè 30 ÁÒãËéáÅéÇÍèÒ¤ÃѺ
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#6
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#5
- -" ¼Á¤§ãªé¤ÓÊ×èͤÇÒÁËÒÂäÁè´ÕÊԹР|
#7
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ÍëÍ à¢éÒã¨áÅéǤÃѺ $a_n$ ä´éà·èÒ¹Ñé¹áËÅФÃѺ á·¹ n=30 仡ç¨Ðä´é $a_{30}=\frac{57}{88}$ Ẻ·Õè⨷ÂìºÍ¡ áÅéǵéͧ¡Òà $a_{24}$ àÃÒ¡çá·¹ $n=24$ ÍèÒ¤ÃѺ ¢Íâ·É¤ÃѺ
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#8
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⨷Âì·Ó¹Í§¹Õéà¨Íºèͨѧ
¶éÒÊÁÁµÔ⨷ÂìÍÍ¡ÁÒẺ¹Õé ¨§ËҨӹǹÊÁÒªÔ¡¢Í§à«µ $\left\{3,5,7,11,...,43\right\}$ ¨ÐµÍºÂÑ§ä§´Õ - -" |
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