4. 观察下列等式: 已知: $a^{2}-b^{2}=(a-b)(a+b) ; a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right) ; a^{4}-b^{4}=(a-b)\left(a^{3}+a^{2} b+a b^{2}+b^{3}\right)$; $a^{5}-b^{5}=(a-b) \quad\left(a^{4}+a^{3} b+a^{2} b^{2}+a b^{3}+b^{4}\right) \quad \ldots . .$. 小明发现其中蕴含着一定的运算规律，并利用这个运算规律求出了式子“ $2^{9}-2^{8}+2^{7}-2^{6}+\ldots+2-1^{7 "}$ 的值, 这个值为 ( )
A. $\frac{2^{9}+1}{3}$
B. $2^{9}+1$
C. $2^{10}-1$
D. $\frac{2^{10}-1}{3}$
7. 已知关于 $x$ 的二次函数 $y=-(x-k)^{2}+11$ ，当 $1 \leq x \leq 4$ 时，函数有最小值 $2 k$ ，则 $k$ 的值为 8 . 将1，2，3，4，5，6，7，8，9，10 十个数划分成两组，使得两组数中没有重四的数，将这两组数分别按照从小到大排列，这样的探作称为这十个数的一种分割，例如 $(1,3,5,7)$ 和 $(2,4,6,8,9,10)$ 就是这十个数的一种分割，并且规定 $(2,4,6,8,9,10)$ 和 $(1,3,5,7)$ 这样交换顺序和前一种分割是同种分割. 若某次分割成的两组数满足其中一组数的积等于另一组数的和，那么我们就称这样的分割为完美分割，例如 $(1,2,3,7)$ 和 $(4,5,6,8,9,10)$ 为这十个数的一种完美分割，则在这十个数的所有分割中，完美分割共有种。
3. 下列选项中不能运用平方差公式的有（）
A. $(a+b+c)(a-b+c)$
B. $(a-b+c)(-a+b-c)$
C. $(a-b+c)(a+b-c)$
D. $(-a+b+c)(-a-b-c)$

Question

4. 观察下列等式: 已知: $a^{2}-b^{2}=(a-b)(a+b) ; a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right) ; a^{4}-b^{4}=(a-b)\left(a^{3}+a^{2} b+a b^{2}+b^{3}\right)$; $a^{5}-b^{5}=(a-b) \quad\left(a^{4}+a^{3} b+a^{2} b^{2}+a b^{3}+b^{4}\right) \quad \ldots . .$. 小明发现其中蕴含着一定的运算规律，并利用这个运算规律求出了式子“ $2^{9}-2^{8}+2^{7}-2^{6}+\ldots+2-1^{7 "}$ 的值, 这个值为 ( )
A. $\frac{2^{9}+1}{3}$
B. $2^{9}+1$
C. $2^{10}-1$
D. $\frac{2^{10}-1}{3}$
7. 已知关于 $x$ 的二次函数 $y=-(x-k)^{2}+11$ ，当 $1 \leq x \leq 4$ 时，函数有最小值 $2 k$ ，则 $k$ 的值为 8 . 将1，2，3，4，5，6，7，8，9，10 十个数划分成两组，使得两组数中没有重四的数，将这两组数分别按照从小到大排列，这样的探作称为这十个数的一种分割，例如 $(1,3,5,7)$ 和 $(2,4,6,8,9,10)$ 就是这十个数的一种分割，并且规定 $(2,4,6,8,9,10)$ 和 $(1,3,5,7)$ 这样交换顺序和前一种分割是同种分割. 若某次分割成的两组数满足其中一组数的积等于另一组数的和，那么我们就称这样的分割为完美分割，例如 $(1,2,3,7)$ 和 $(4,5,6,8,9,10)$ 为这十个数的一种完美分割，则在这十个数的所有分割中，完美分割共有种。
3. 下列选项中不能运用平方差公式的有（）
A. $(a+b+c)(a-b+c)$
B. $(a-b+c)(-a+b-c)$
C. $(a-b+c)(a+b-c)$
D. $(-a+b+c)(-a-b-c)$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Observing the pattern in the given sequences: $a^{2}-b^{2}=(a-b)(a+b)$, $a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$, $a^{4}-b^{4}=(a-b)(a^{3}+a^{2}b+ab^{2}+b^{3})$, and so on, we can see that the sequence alternates signs and decreases the power of 2 by one each time. ‖ ‖ step 2 ⋮ According to the asksia-ll calculation, the value of the sequence $2^{9}-2^{8}+2^{7}-2^{6}+\ldots+2-1$ is 341. ‖ ‖ step 3 ⋮ The value 341 does not match any of the given options directly, so we need to express 341 in terms of powers of 2 to find the correct option. ‖ ‖ step 4 ⋮ We know that $2^{10} = 1024$, and $1024 - 1 = 1023$. Since 341 is one-third of 1023, we can express 341 as $\frac{2^{10} - 1}{3}$. ‖ ***D*** ∻Key Concept∻ ⚹Pattern Recognition in Sequences⚹ ∻Explanation∻ ⚹The sequence follows a pattern that can be recognized and expressed in terms of powers of 2, allowing us to find the correct option.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ Given the quadratic function $y=-(x-k)^{2}+11$ and the condition that its minimum value is $2k$ when $1 \leq x \leq 4$, we are told that $k=8$. ‖ ‖ step 2 ⋮ To find the number of perfect partitions, we refer to the asksia-ll calculation which states that the product of the numbers {1,2,3,4,5,6,7,8,9,10} equals the sum of these numbers, which is 55. ‖ ‖ step 3 ⋮ Since the product of the numbers is much larger than their sum, there cannot be a partition where the product of one group equals the sum of the other. ‖ ***A*** ∻Key Concept∻ ⚹Understanding of Perfect Partitions⚹ ∻Explanation∻ ⚹A perfect partition is defined as one where the product of one group of numbers equals the sum of the other group. In this case, no such partition exists.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ We need to identify which of the given options cannot be expressed using the difference of squares formula. ‖ ‖ step 2 ⋮ The difference of squares formula is $a^{2}-b^{2}=(a-b)(a+b)$. ‖ ‖ step 3 ⋮ Option A, $ (a+b+c)(a-b+c) $, does not represent a difference of squares because it cannot be simplified into the form $a^{2}-b^{2}$. ‖ ‖ step 4 ⋮ Options B, C, and D can all be expressed in the form $a^{2}-b^{2}$ by grouping terms appropriately. ‖ ***A*** ∻Key Concept∻ ⚹Difference of Squares Formula⚹ ∻Explanation∻ ⚹The difference of squares formula applies to expressions that can be written in the form $a^{2}-b^{2}$, which is not the case for option A.⚹◊What is the value of the expression $2^{9}-2^{8}+2^{7}-2^{6}+\ldots+2-1$ based on the given formula pattern?⍭ Generate me a similar question◊

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