gwd 26-29

bettyofarim

Q29:

If the sequence x₁, x_2, x_{3, …,} x_n, … is such that x₁ = 3 and x_n+1
= 2x_n – 1 for n ≥ 1, then x₂₀ – x₁₉ =
A. 2¹⁹
B. 2²⁰
C. 2²¹
D. 2²⁰ - 1
E. 2²¹ - 1
我怎么都算不出答案，谢谢各位大侠

daoying

X20 - X19
= (2 * X19 - 1) - X19
= X19 - 1
= (2 * X18 - 1) - 1
= 2 * (2 * X17 - 1) - 2
= 2 * (2 * (2 * X16 - 1) - 1) - 2
= 2 * 2 * 2 * X16 - (2 * 2 - 2 - 1) - 1
= 2^18 * X1 - (2^17 - 2^16 - ... - 2^2 - 2^1 - 2^0) - 1
= 2^18 * 3 - sum of(2^k) where k goes from 0 to 17 (see note) - 1
= 2^18 * 3 - 2^0 * (2^18 - 1) / (2 - 1) - 1
= 2^18 * 3 - 1 * 2^18 + 1 - 1
= 2^18 * 2
= 2^19

Note: sum of (x^k) where k goes from y to z = x^y * (x^(z - y + 1) - 1) / (x - 1)

A

longxiaomeim

ding !!!!!!!!!!!