等差数列an的前n项和为Sn,已知a2=3,Sn-S(n-3)=243,Sn=420,则n=?

Question

等差数列an的前n项和为Sn,已知a2=3,Sn-S(n-3)=243,Sn=420,则n=?

Answer 1

s(n)-s(n-3)
=a(n)+a(n-1)+a(n-2)
=[a(1)+(n-1)d]+[a(1)+(n-2)d]+[a(1)+(n-3)d]
=3a(1)+3(n-2)d=243
a(1)+(n-2)d=81
a(2)=a(1)+d=3
s(n)=[a(1)+a(n)]n/2
=[2a(1)+(n-1)d]n/2
=420
a(1)+(n-2)d=81
a(1)+d=3
[2a(1)+(n-1)d]n/2=420
(n-3)d=78
d=78/(n-3)
a(1)+d=a(1)+78/(n-3)=3
a(1)=3-78/(n-3)
[6-156/(n-3)+(n-1)*78/(n-3)]*n/2=420
[6(n-3)-156+78(n-1)]*n/(n-3)=840
(84n-252)n/(n-3)=840
(n-3)n/(n-3)=10
n=10
d=78/7
a(1)=-57/7