设F1,F2分别是椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点,过F1斜率为1的直线l与E相交于A,

|AF2|,|AB|,|BF2|成等差数列则：2AB=AF2+BF2即：2AF1+2BF1=AF2+BF2 ①设A(x1,y1),B(x2,y2)则由焦半径公式：AF1=a+ex1,AF2=a-ex1,BF1=a+ex2,BF2=a-ex2代入①式得：4a+2e(x1+x2)=2a-e(x1+x2)3e(x1+x2)=-2a 直线L：y=x+c代入椭圆得：x²/a²+(x+c)²/b²=1即：(1/a²+1/b²)x²+2cx/b²+c²/b²-1=0由韦达定理：x1+x2=-(2c/b²)/(1/a²+1/b²)=-2ca²/(a²+b²)代入②得：-6eca²/(a²+b²)=-2a-3ac²/(a²+b²)=-a3c²=a²+b²3c²=a²+a²-c²2c²=a²e²=1/2所以,离心率e=√2/2