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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 :haha: |
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) $ $=1+2+2^2+2^3+...+2^9$ $=2^{10}-1$ |
¨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 |
¢éÍáá¹ÕèÍÂÒ¡ãËéµÍºÇèÒ $a_n=\dfrac{2n-3}{3n-2}$ ËÃ×Íà»ÅèÒ¤ÃѺ
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