CMR1×2-1/2!+2×3-1/2!+3×4-1/4!+...+2023×2024/2024!<2
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Đặt A = 1 + 2 + 3 + 4 + ... + 2023
Tổng có 2023 - 1 + 1 số hạng
A = (2023 + 1) × 2023 : 2
= 2047276
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Đặt B = 20 + 21 + 22 + ... + 2024
Tổng có: 2024 - 20 + 1 = 2005 số hạng
B = (2024 + 20) × 2005 : 2
= 2049110
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Đặt C = 2 + 4 + 6 + ... + 2024
Tổng có (2024 - 2) : 2 + 1 = 1012 số hạng
C = (2024 + 2) × 1012 : 2
= 1025156
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Đặt D = 1 + 2 + 4 + 8 + 16 + ... + 8192
2 × D = 2 + 4 + 8 + 16 + 32 + ... + 16384
2 × D - D = (2 + 4 + 8 + 16 + 32 + ... + 16384) - (1 + 2 + 4 + 8 + 16 + ... + 8192)
= 16384 - 1
= 16383
Vậy D = 16383
\(a,A=1+2+3+4+5..+2023\)
Số số hạng:
\(\left(2023-1\right):1+1=2023\)
Tổng :
\(\dfrac{\left(2023+1\right).2023}{2}=2047276\)
\(b,20+21+22+..+2024\)
Số số hạng:
\(\left(2024-20\right):1+1=2005\)
Tổng:
\(\dfrac{\left(2024+20\right).2005}{2}=2049110\)
\(c,2+4+6+..+2024\)
Số số hạng:
\(\left(2024-2\right):2+1=1012\)
Tổng:
\(\dfrac{\left(2024+2\right).1012}{2}=1025156\)
\(\dfrac{1}{2}\times\dfrac{2}{3}\times\dfrac{3}{4}\times...\times\dfrac{2023}{2024}\\ =\dfrac{1\times2\times3\times...\times2023}{2\times3\times4\times...\times2024}\\ =\dfrac{1}{2024}\)
\(A=\dfrac{1}{2}-\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^3-\left(\dfrac{1}{2}\right)^4+...+\left(\dfrac{1}{2}\right)^{2023}-\left(\dfrac{1}{2}\right)^{2024}\)
\(A=\dfrac{2}{2^2}-\dfrac{1}{2^2}+\dfrac{2}{2^4}-\dfrac{1}{2^4}+...+\dfrac{2}{2^{2024}}-\dfrac{1}{2^{2024}}\)
\(A=\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+...+\dfrac{1}{2^{2024}}\)
\(A=\dfrac{2^{2022}}{2^{2024}}+\dfrac{2^{2020}}{2^{2024}}+\dfrac{2^{2018}}{2^{2024}}+...+\dfrac{1}{2^{2024}}\)
\(2^2A=\dfrac{2^{2024}}{2^{2024}}+\dfrac{2^{2022}}{2^{2024}}+\dfrac{2^{2020}}{2^{2024}}+...+\dfrac{2^2}{2^{2024}}\)
\(\Rightarrow4A-A=3A=1-\dfrac{2}{2^{2024}}-\dfrac{1}{2^{2024}}\)
\(3A=1-\left(\dfrac{2}{2^{2024}}+\dfrac{1}{2^{2024}}\right)\)
\(3A=1-\dfrac{3}{2^{2024}}\)
\(A=\dfrac{1-\dfrac{3}{2^{2024}}}{3}\)
\(A=\dfrac{3\left(\dfrac{1}{3}-\dfrac{1}{2^{2024}}\right)}{3}\)
\(A=\dfrac{1}{3}-\dfrac{1}{2^{2024}}\)
1-2+3-4+5-6+7-8+...+2023-2024
=(1−2)+(3−4)+(5−6)+(7−8)+....+(2023−2024)=(1−2)+(3−4)+(5−6)+(7−8)+....+(2023−2024)
=−1+(−1)+(−1)+(−1)+...+(−1)=−1+(−1)+(−1)+(−1)+...+(−1)
=−1.1012=−1.1012
=−1012=−1012
1-2+3-4+5-6+ ... +2023-2024
= (-1) + (-1) + ... + (-1) (1012 số)
= (-1).1012
= -1012
\(\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{2023}\right)\left(1-\dfrac{1}{2024}\right)\)
=\(=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}...\dfrac{2022}{2023}.\dfrac{2023}{2024}=\dfrac{1}{2024}\)
\(\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{2024}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot\dfrac{3}{4}\cdot\dfrac{4}{5}\cdot...\cdot\dfrac{2023}{2024}\)
\(=\dfrac{1\cdot2\cdot3\cdot4\cdot...\cdot2023}{2\cdot3\cdot4\cdot5\cdot...\cdot2024}\)
\(=\dfrac{1}{2024}\)
42 : x + 36 : x = 6
TH1
42:x=6
x= 42 :6
X= 7
TH 2
36:x = 6
X = 36: 6
X= 6