Cho A = 1 - 3/4 + (3/4)2 - (3/4)3 + (3/4)4 + ... - (3/4)2023 Chứng minh rằng A ko có phải là số nguyên
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Dễ thấy A > 1
Ta có:
\(A=\frac{1}{1^2}+\frac{1}{2^3}+...+\frac{1}{2018^{2019}}\)
\(< \frac{1}{1^2}+\frac{1}{2^2}+...+\frac{1}{2018^2}< 1+\frac{1}{1\cdot2}+...+\frac{1}{2017\cdot2018}\)
\(=1+1-\frac{1}{2}+...+\frac{1}{2018}=2-\frac{1}{2018}< 2\)
Vì \(1< A< 2\) nên A không nguyên
\(A=\dfrac{3}{2^2}+\dfrac{8}{3^2}+\dfrac{15}{4^2}+...+\dfrac{2023^2-1}{2023^2}\)
\(A=\dfrac{2^2-1}{2^2}+\dfrac{3^2-1}{3^2}+\dfrac{4^2-1}{4^2}+...+\dfrac{2023^2-1}{2023^2}\)
\(A=1-\dfrac{1}{2^2}+1-\dfrac{1}{3^2}+1-\dfrac{1}{4^2}+...+1-\dfrac{1}{2023^2}\)
\(A=(1+1+1+...+1)-(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+..+\dfrac{1}{2023^2})\)
Tổng số hạng của 2 ngoặc trên bằng nhau và =(2023-2):1+1=2022(số hạng)
\(A=2022-(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{2023^2})\)
Ta thấy:
\(0<\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{2023^2}<\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+..+\dfrac{1}{2022.2023}\)
Ta có
\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+..+\dfrac{1}{2022.2023}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+..+\dfrac{1}{2022}-\dfrac{1}{2023}\)
\(=1-\dfrac{1}{2023}<1\)
Do đó,2021<A<2022
Vậy giá trị của A không phải 1 số tự nhiên(đpcm)
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1-122+1-132+1-142+....+1-120232"" id="MathJax-Element-2-Frame" role="presentation" tabindex="0" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">A=1−122+1−132+1−1(2+....+1)120232�=1-122+1-132+1-142+....+1-1202321+12+13+...+122023−1
2022-(122+132+142+...+120232)"" id="MathJax-Element-3-Frame" role="presentation" tabindex="0" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">A=2022−(122+132+142+...+120232)�=2022-(122+132+142+...+120232)A
122+132+142+.... <20232
M=(1/5+1/5^2+1/5^3+...+1/5^2023) + 1/5x(1/5+1/5^2+1/5^3+...+1/5^2022) + ... + 1/5^2021x(1/5+1/5^2) + 1/5^2022x1/5
Xét biểu thức N=1/5+1/5^2+1/5^3 + ... + 1/5^k (K>0, k thuộc Z)
=> 5N=1+1/5+1/5^2+1/5^3+...+1/5^(k-1)
=> 4N= 5N - N =1 - 1/5^k
=> 1/5+1/5^2+1/5^3 + ... + 1/5^k = 1/4x(1-1/5^k)
Thay vào biểu thức M, ta có:
M= 1/4x(1-1/5^2023) + 1/5x1/4x(1-1/5^2022) + ... + 1/5^2021x1/4x(1-1/5^2) + 1/5^2022x1/4x(1-1/5)
=> 4M = (1+1/5+1/5^2+...+1/5^2022) - 2023/5^2023
=> 4M = 5/4x(1-1/5^2023)-2023/5^2023 < 5/4
=> M < 5/16 < 1/3
Vậy M < 1/3 [ vượt chỉ tiêu nhé =)) ]