AC是正方形ABCD的对角线,BE1⊥AC,E1F1⊥AB,F1E2⊥AC,E2F2⊥AB,F2E3⊥AC.求AE3：AB；求AEn：AB

Question

AC是正方形ABCD的对角线,BE1⊥AC,E1F1⊥AB,F1E2⊥AC,E2F2⊥AB,F2E3⊥AC.求AE3：AB；求AEn：AB

Answer 1

(1)∵AE1⊥AC;AB=BC.
∴AE1=CE1=AC/2=(√2AB)/2.
同理可证:
AE2=E1E2,即AE2=(1/2)AE1;
AE3=(1/2)AE2=(1/2)²AE1=(1/4)(√2AB)/2=(√2/8)AB.
故AE3:AB=√2:8.
(2)同样的道理可知:
AE4=(1/2)³(√2AB/2);
……
AEn=[(1/2)^(n-1)](√2AB/2)
故AEn:AB=√2:[2^(n-1)]*2=√2:(2^n)