of n−1 real numbers. Define A1(S)=A(S) and, for each integer m,2≤m≤n− 1 , define Am(S)=A(Am−1(S)). Suppose x>0, and let S=(1,x,x2,…,x100). If A100(S)=(1/250), then what is x?
Answer Choices:
A. 1−22​​
B. 2​−1
C. 21​
D. 2−2​
E. 22​​ Solution:
For every sequence S=(a1​,a2​,…,an​) of at least three terms,
Thus for m=1 and 2 , the coefficients of the terms in the numerator of Am(S) are the binomial coefficients (0m​),(1m​),…,(mm​), and the denominator is 2m. Because (rm​)+(r+1m​)=(r+1m+1​) for all integers r≥0, the coefficients of the terms in the numerators of Am+1(S) are (0m+1​),(1m+1​),…,(m+1m+1​) for 2≤m≤n−2. The definition implies that the denominator of each term in Am+1(S) is 2m+1. For the given sequence, the sole term in A100(S) is