数列an＞0,若a1³＋a2³＋a3³＋……＋an³＝（a1＋a2＋a3＋……＋an）²

＝（a1＋a2＋a3＋……＋an）²若a1³＋a2³＋a3³＋……＋an³

＋……＋a(n-1)³＋a3³＋……＋an³＝（a1＋a2＋a3＋……＋an）²a1³＋a2³＋a2³＋a3³＝（a1＋a2＋a3＋……＋an-1）²

相减得
an³ = an * [2(a1＋a2＋a3＋……＋an) - an] = an * (2Sn -an)
所以
an² = 2Sn-an
Sn = (an²+an)/2
Sn-1 = (an-1²+an-1)/2
相减得
2an = an² + an - a(n-1)² - a(n-1)
所以(an+a(n-1))(an-a(n-1)) - (an+a(n-1)) = 0
由于an>

a1³

a1³＋a2³＋a3³＋……＋an³＝（a1＋a2＋a3＋……＋an）² a1³＋a2³＋a3³＋……＋a(n-1)³＝（a1＋a2＋a3＋……＋an-1）² 相减得 an³ = an * [2(a1＋a2＋a3＋……＋an) - an] = an * (2sn -an) 所以 an² = 2sn-an sn = (an²+an)/2 sn-1 = (an-1²+an-1)/2 相减得 2an = an² + an - a(n-1)² - a(n-1) 所以(an+a(n-1))(an-a(n-1)) - (an+a(n-1)) = 0 由于an>0 所以an+a(n-1) ≠ 0 所以 an = a(n-1) + 1 又a1 = 1 所以an = n