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1题目设a_1,a_2,a_3,a_4是各项为正数且公差为d(d≠0)的等差数列。(Ⅰ)证明:2~(a_1),2~(a_2),2~(a_3),2~(a_4)依次构成等比数列;(Ⅱ)是否存在a_1、d,使得a_1,a_2~2,a_3~3,a_4~4依次构成等比数列,并说明理由;(Ⅲ)是否存在a_1、d及正整数n、k,使得a_1~n,a_2~(n+k),a_3~(n+2k),a_4~(n+3k)依次构成等比数列,并说明理由。2赏析2.1选材——主干知识,交汇自然这是文理科合卷的最后一题,该题涉及中学数学的很多主干知识,比如等差数列、等比数列、指数运算与对数运算、一元二次函数、对数函数、导数、方程的
1 The subject set a_1, a_2, a_3, a_4 is a positive number and tolerance d (d ≠ 0) of the arithmetic sequence. (A), 2 ~ (a_2), 2 ~ (a_3), 2 ~ (a_4) successively form the geometric sequence; (Ⅱ) whether there exists a_1, d such that a_1, a_2 ~ 2, a_3 ~ 3, a_4 ~ 4 constitute the geometric sequence, and explain the reasons; (Ⅲ) there is a_1, d and positive integer n, k, so that a_1 ~ n, a_2 ~ (n + k), a_3 ~ (n + 2k), a_4 ~ (n + 3k) constitute the geometric sequence, and explain the reasons. 2 Appreciation 2.1 Selection of materials - the backbone of knowledge, to meet the natural This is the last question of science and engineering co-curricular, the subject involves many of the backbone of high school mathematics knowledge, such as arithmetic sequence, arithmetic sequence, index and logarithm operation, Functions, logarithms, derivatives, equations