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若数列﹛an﹜满足条件：存在正整数k,使得an+k+an-k=2an若﹛an﹜既是2级等差数列又是3级等差数列,证明它是等差数列
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若数列﹛an﹜满足条件：存在正整数k,使得an+k + an-k=2an 若﹛an﹜既是2级等差数列又是3级等差数列,证明它是等差数列
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a(n+3)+a(n-3)=2a(n), a(n+6)+a(n)=2a(n+3),
a(n+2)+a(n-2)=2a(n), a(n+4)+a(n)=2a(n+2).
a(n+6)-a(n+3) = a(n+3) - a(n),
a(n+4)-a(n+2) = a(n+2) - a(n).
a(n+3)-a(n) = a(n+6) - a(n+3) = a(n+2+4)-a(n+2+2) + a(n+4) - a(n+3)= a(n+2+2) - a(n+2) + a(n+4) - a(n+3) = 2a(n+4)- a(n+2)-a(n+3),
2a(n+4)-2a(n+3) = a(n+2)-a(n),
2a(n+4) - 2a(n+2) = 2a(n+3) - a(n+2)-a(n).
2a(n+2)-2a(n) = 2a(n+4) - 2a(n+2) = 2a(n+3) - a(n+2) - a(n),
0 = 2a(n+3) - 3a(n+2) + a(n).
2a(n+3) - a(n+2) - a(n+1) = 2a(n+2) - a(n+1) - a(n),
{2a(n+2)-a(n+1)-a(n)}是首项为b = 2a(3)-a(2)-a(1),的常数数列.
2a(n+2) - a(n+1) - a(n) = 2a(3)-a(2)-a(1) = b
2a(n+2) - 2a(n+1) + a(n+1) - a(n) = b,
2[a(n+2) - a(n+1)] = -[a(n+1)-a(n)] + b = -[a(n+1)-a(n)] +2b/3 + b/3
2[a(n+2) - a(n+1)-b/3] = -[a(n+1)-a(n)-b/3],
a(n+2) - a(n+1) - b/3 = (-1/2)[a(n+1)-a(n)-b/3],
{a(n+1)-a(n) - b/3}是首项为a(2)-a(1)-b/3 = a(2)-a(1)-[2a(3)-a(2)-a(1)]/3 = [-2a(3)+4a(2)-2a(1)]/3, 公比为(-1/2)的等比数列.
a(n+1)-a(n) -b/3 = [-2a(3)+4a(2)-2a(1)]/3*(-1/2)^(n-1).
2a(n+1)-2a(n) - 2b/3 = [-4a(3)+8a(2)-4a(1)]/3*(-1/2)^(n-1).
2a(n+2)-a(n+1)-a(n)=b.
2a(n+2)+a(n+1) = 2a(n+1) + a(n) + b,
{2a(n+1)+a(n)}是首项为2a(2)+a(1),公差为b的等差数列.
2a(n+1)+a(n) = [2a(2)+a(1)] + (n-1)b.
3a(n) + 2b/3 = [2a(n+1)+a(n)]-[2a(n+1)-2a(n)-2b/3] = 2a(2)+a(1) + (n-1)b + [4a(3)-8a(2)+4a(1)]/3*(-1/2)^(n-1)
3a(n) = 2a(2) + a(1) - 2b/3 + (n-1)b + (4/3)[a(3)-2a(2)+a(1)](-1/2)^(n-1)
= 2a(2) + a(1) - 2[2a(3)-a(2)-a(1)]/3 + (n-1)b + (4/3)[a(3)-2a(2)+a(1)](-1/2)^(n-1)
= [-4a(3) + 8a(2) + 5a(1)]/3 + (n-1)b + (4/3)[a(3)-2a(2)+a(1)](-1/2)^(n-1)
a(n) = [-4a(3)+8a(2)+5a(1)]/9 + (n-1)[2a(3)-a(2)-a(1)]/3 + (4/9)[a(3)-2a(2)+a(1)](-1/2)^(n-1),
当且仅当 a(3)-2a(2)+a(1)=0时,a(n) = [-4a(3)+8a(2)+5a(1)]/9 + (n-1)[2a(3)-a(2)-a(1)]/3 ,
为首项为 [-4a(3)+8a(2)+5a(1)]/9, 公差为[2a(3)-a(2)-a(1)]/3的等差数列.
而且,a(3)-2a(2)+a(1)=0时,[-4a(3)+8a(2)+5a(1)]/9 = [-4a(3)+8a(2)-4a(1)+9a(1)]/9 = a(1),
[2a(3)-a(2)-a(1)]/3 = [2a(3)-4a(2)+2a(1)+3a(2)-3a(1)]/3 = a(2)-a(1),
a(3)-2a(2)+a(1)=0时,a(n)为首项为a(1),公差为a(2)-a(1)的等差数列.
a(n+2)+a(n-2)=2a(n), a(n+4)+a(n)=2a(n+2).
a(n+6)-a(n+3) = a(n+3) - a(n),
a(n+4)-a(n+2) = a(n+2) - a(n).
a(n+3)-a(n) = a(n+6) - a(n+3) = a(n+2+4)-a(n+2+2) + a(n+4) - a(n+3)= a(n+2+2) - a(n+2) + a(n+4) - a(n+3) = 2a(n+4)- a(n+2)-a(n+3),
2a(n+4)-2a(n+3) = a(n+2)-a(n),
2a(n+4) - 2a(n+2) = 2a(n+3) - a(n+2)-a(n).
2a(n+2)-2a(n) = 2a(n+4) - 2a(n+2) = 2a(n+3) - a(n+2) - a(n),
0 = 2a(n+3) - 3a(n+2) + a(n).
2a(n+3) - a(n+2) - a(n+1) = 2a(n+2) - a(n+1) - a(n),
{2a(n+2)-a(n+1)-a(n)}是首项为b = 2a(3)-a(2)-a(1),的常数数列.
2a(n+2) - a(n+1) - a(n) = 2a(3)-a(2)-a(1) = b
2a(n+2) - 2a(n+1) + a(n+1) - a(n) = b,
2[a(n+2) - a(n+1)] = -[a(n+1)-a(n)] + b = -[a(n+1)-a(n)] +2b/3 + b/3
2[a(n+2) - a(n+1)-b/3] = -[a(n+1)-a(n)-b/3],
a(n+2) - a(n+1) - b/3 = (-1/2)[a(n+1)-a(n)-b/3],
{a(n+1)-a(n) - b/3}是首项为a(2)-a(1)-b/3 = a(2)-a(1)-[2a(3)-a(2)-a(1)]/3 = [-2a(3)+4a(2)-2a(1)]/3, 公比为(-1/2)的等比数列.
a(n+1)-a(n) -b/3 = [-2a(3)+4a(2)-2a(1)]/3*(-1/2)^(n-1).
2a(n+1)-2a(n) - 2b/3 = [-4a(3)+8a(2)-4a(1)]/3*(-1/2)^(n-1).
2a(n+2)-a(n+1)-a(n)=b.
2a(n+2)+a(n+1) = 2a(n+1) + a(n) + b,
{2a(n+1)+a(n)}是首项为2a(2)+a(1),公差为b的等差数列.
2a(n+1)+a(n) = [2a(2)+a(1)] + (n-1)b.
3a(n) + 2b/3 = [2a(n+1)+a(n)]-[2a(n+1)-2a(n)-2b/3] = 2a(2)+a(1) + (n-1)b + [4a(3)-8a(2)+4a(1)]/3*(-1/2)^(n-1)
3a(n) = 2a(2) + a(1) - 2b/3 + (n-1)b + (4/3)[a(3)-2a(2)+a(1)](-1/2)^(n-1)
= 2a(2) + a(1) - 2[2a(3)-a(2)-a(1)]/3 + (n-1)b + (4/3)[a(3)-2a(2)+a(1)](-1/2)^(n-1)
= [-4a(3) + 8a(2) + 5a(1)]/3 + (n-1)b + (4/3)[a(3)-2a(2)+a(1)](-1/2)^(n-1)
a(n) = [-4a(3)+8a(2)+5a(1)]/9 + (n-1)[2a(3)-a(2)-a(1)]/3 + (4/9)[a(3)-2a(2)+a(1)](-1/2)^(n-1),
当且仅当 a(3)-2a(2)+a(1)=0时,a(n) = [-4a(3)+8a(2)+5a(1)]/9 + (n-1)[2a(3)-a(2)-a(1)]/3 ,
为首项为 [-4a(3)+8a(2)+5a(1)]/9, 公差为[2a(3)-a(2)-a(1)]/3的等差数列.
而且,a(3)-2a(2)+a(1)=0时,[-4a(3)+8a(2)+5a(1)]/9 = [-4a(3)+8a(2)-4a(1)+9a(1)]/9 = a(1),
[2a(3)-a(2)-a(1)]/3 = [2a(3)-4a(2)+2a(1)+3a(2)-3a(1)]/3 = a(2)-a(1),
a(3)-2a(2)+a(1)=0时,a(n)为首项为a(1),公差为a(2)-a(1)的等差数列.
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