Quickly write out the first three terms:
X1 = 3
X2 = 2*X1=2*3
X3=2*X2=2(2*3)=2^(2)*3
Immediately you should see that XN=2^(n-1)*3. Then X20-X19=2^(19)*3-2^(18)*3=3*2^(18)[2-1]=3*2^(18)=X19.
If the sequence X 1 (X one), X 2 (X two), X 3 (X three), .....Xn is such that X 1= 3 and X n+1 = 2 Xn -1 for n=1 , then X 20 - X 19 =
A) 2 ^ 19
B) 2 ^ 20
C) 2 ^ 21
D) (2 ^ 20) -1
E) (2 ^ 21) -1
Can some one please explain how this problem can be solved without making the lengthy calculations.
Xn+1 = (2Xn) - 1 or Xn = (Xn-1) - 1
X1 = 3 = 2^1 + 1
X2 = 2*(3) - 1 = 5 = 2^2 + 1
X3 = 2*5 - 1 = 9 = 2^3 + 1
So, it follows that the nth term of the sequence is obviously Xn = 2^n + 1
Therefore, X^20 - X^19 = 2^20 + 1 - 2^19 - 1 = 2^20 - 2^19 = 2^19 * (2-1) = 2^19.
(Choice A is the answer)
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