猫猫强
幼苗
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S1=a1=[8/9],S2=a1+a2=[24/25],S3的=S2 +a3=[48/49].
猜测 Sn =
(2n+1)2−1
(2n+1)2.
证明:①当n=1时,由以上可知,猜测成立.
②假设n=k时,猜测成立,即 SK=
(2k+1)2−1
(2k+1)2.
则n=k+1时,SK+1=SK+ak+1=
(2k+1)2−1
(2k+1)2+
8(k+1)
[4(k+1)2−1]2
=
(2k+1)2−1
(2k+1)2+
8(k+1)
(2k+1)2(2k+3)2=
[(2k+1)2−1](2k+3)2+8(k+1)
(2k+1)2(2k+3)2
=
(2k+3)2−1
(2k+1)2(2k+3)2=
[2(k+1)+1 ]2−1
(2k+1)2(2k+3)2.
故当n=k+1时,猜测仍然成立.
综合①②可得,猜测对任意的正整数都成立.
1年前
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