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等差和問題

等差和問題

請教各位老師,此題怎解,謝謝。
(無答案)
9.
假設a1a2a3a11為等差數列,公差為2 ,且總和為445 ,試求
a22a1a3+a23a2a4+a24a3a5+a25a4a6+a26a5a7+a27a6a8+a28a7a9+a29a8a10+a210a9a11之值。

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試算了一下,參考看看
a2k+1akak+2=(ak+d)2(ak)(ak+2d)=a2k+2akda2k+2akd+d2=1+d2a2k+2akd=1+d2(ak)(ak+2d)=1+d2211ak1ak+2d 

9k=1a2k+1akak+2=9k=11+d2211ak1ak+2d=9k=11+d2219k=11ak1ak+2d 

=9+(2)2211a11a3+1a21a4++1a91a11=9+1a1+1a21a101a11=9+22 

S11=445d=2 

a1=4552a2=4542a10=45+42a11=45+52 

1a1+1a21a101a11=1a11a11+1a21a10=a1a11a11a1+a2a10a10a2 

=(4552)(45+52)(45+52)(4552)+(4542)(45+42)(45+42)(4542)=1028050+828032=3010+8482=22 

看來豈是尋常色   濃淡由他冰雪中

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