Question

已知数列{an}满足a1=1,a2=-13,a(n+2)-2(an+1)+an=2n-6 1)设bn=A(n+1)-An,求数列{bn}的通项公式

Answer 1

b1=a2-a1
=-13-1
=-14

bn=a(n+1)-an
a(n+2)-2a(n+1)+an=2n-6
a(n+2)-a(n+1)-a(n+1)+an=2n-6
a(n+2)-a(n+1)-[a(n+1)+an]=2n-6
b(n+1)-bn=2n-6
所以bn-b(n-1)=2(n-1)-6
bn-b(n-1)=2(n-1)-6
...........
b3-b2=2*2-6
b2-b1=2*1-6
以上等式相加得
bn-b1=2*1-6+2*2-6+.........+2(n-1)-6
bn-b1=2*(1+2+3+......+n-1)-6(n-1)
bn-b1=n(n-1)-6n+6
bn-(-14)=n^2-n-6n+6
bn+14=n^2-7n+6
bn=n^2-7n-8