trustkip wrote:Does anyone have any idea how to solve the following problem?
If the sequence x_1, x_2, x_3, ..., x_n, ... is such that x_1 = 3 and x_(n+1) = 2(x_n} - 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
Write it out and LOOK FOR A PATTERN.
The formula in red indicates that every term is 1 less than twice the preceding term.
Thus:
x_2 = 2(x_1) - 1 = 2*3 - 1 = 5.
x_3 = 2(x_2) - 1 = 2*5 - 1 = 9.
x_4 = 2(x_3) - 1 = 2*9 - 1 = 17.
Since the answer choices all include a power of 2, try to express each term in terms of a POWER OF 2.
The results above -- 5, 9, 17 -- imply the pattern indicated by the values in red:
x_
2 = 2^
2 + 1.
x_
3 = 2^
3 + 1.
x_
4 = 2^
4 + 1.
Implication:
x_
20 - x_
19
= (2^
20 + 1) - (2^
19 + 1)
= 2^20 - 2^19
= 2^19(2-1)
= 2^19.
The correct answer is
A.
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