已知函数f(t)对任意函数x,y都有f(x+y)=f(x)+f(y)=3xy(x+y+2)+3,f(1)=1.

1个回答

  • (1):设x=t,y=1则f(t+1)=f(t)+f(1)+3t(t+3)+3

    =f(t)+3t(t+3)+4

    f(t+1)-f(t)=3t(t+3)+4=3t^2+9t+4

    f(2)-f(1)=3*1^2+9*1+4 ,(1)

    f(3)-f(2)=3*2^2+9*2+4 ,(2)

    f(4)-f(3)=3*3^2+9*3+4 ,(3)

    ...

    f(t)-f(t-1)=3*(t-1)^2+9*(t-1)+4 ,(t-1)

    f(t+1)-f(t)=3*t^2+9*t+4 ,(t)

    (1)+(2)+(3)+...+(t-1)+(t)

    得f(t+1)-f(1)=3*(1^2+2^2+3^2+...t^2)+9*(1+2+3+...+t)+4*t

    =t(t+1)(2t+1)/2+9t(t+1)/2+4t

    f(t+1)=(t+1-1)(t+1)(2(t+1)-1)/2+9(t+1-1)(t+1)/2+4(t+1-1)+1

    所以f(t)=(t-1)t(2t-1)/2+9(t-1)t/2+4(t-1)+1

    f(t)=(t-1)t(t+4)+4(t-1)+1

    (2):

    (t-1)t(t+4)+4(t-1)+1=t

    (t-1)t(t+4)+4(t-1)=t-1

    若t1则t^2+4t+3=0

    解得t=-1或t=-3

    即t为-3,-1,1是首项为-3,公差为2的等差数列

    解毕.