Where A0, A 1, A2, ..., an are arithmetic, f(0)=f( 1)= 105, f(- 1)= 15, find the values of n and an.
Analysis: ∫ f (x) = A0+a1x+a2x2+a3x3+...+a (n-1) x (n-1)+anxn.
Where A0, A 1, A2, ..., an are equal, f(0)=f( 1)= 105, f(- 1)= 15.
∴f(0)=a0= 105,f( 1)= 105+a 1+a2+…+an= 105
a0+(a0+d)+(a0+2d)+.....+(A0+nd)= 105(n+ 1)+n(n+ 1)d/2 = 105
= = & gt 105n=-n(n+ 1)d/2
∫N∈N *
∴(n+ 1)d=-2 10( 1)
When n is odd, f (-1) = (A0-a1)+(A2-A3)+...+[A (N-1)-an] = (N+1) d/2.
N is an even number.
That is, f (-1) = A0+(-a1+a2)+(-a3+a4)+...+[-a (n-1)+an].
∫an-a(n- 1)= A0+nd-A0-(n- 1)d = d
∴f(- 1)=a0+nd/2= 105+nd/2= 15(2)
(1)(2) The simultaneous solution is n=6 and d=-30.
An =-75