tính hợp lý nếu có thể A=1.2+2.3+...+(2013.20.19)-1^2+2+3^2+...+2013^2)
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Những câu hỏi liên quan
NL
Nguyễn Lê Phước Thịnh
CTVHS
4 tháng 6
a: \(M=1\cdot2+2\cdot3+\cdots+2020\cdot2021\)
\(=1\left(1+1\right)+2\left(2+1\right)+\cdots+2020\left(2020+1\right)\)
\(=\left(1^2+2^2+\cdots+2020^2\right)+\left(1+2+\cdots+2020\right)\)
\(=\frac{2020\left(2020+1\right)\left(2\cdot2020+1\right)}{6}+\frac{2020\cdot2021}{2}\)
\(=\frac{2020\cdot2021\cdot4041+3\cdot2020\cdot2021}{6}=\frac{2020\cdot2021\cdot4044}{6}\)
\(=2020\cdot2021\cdot674=2751551080\)
b: \(N=1\cdot2\cdot3+2\cdot3\cdot4+\cdots+2019\cdot2020\cdot2021\)
\(=2\left(2-1\right)\left(2+1\right)+3\left(3-1\right)\left(3+1\right)+\cdots+2020\left(2020-1\right)\left(2020+1\right)\)
\(=2\left(2^2-1\right)+3\left(3^2-1\right)+\cdots+2020\left(2020^2-1\right)\)
\(=\left(2^3+3^3+\cdots+2020^3\right)-\left(2+3+\cdots+2020\right)\)
\(=\left(1^3+2^3+\cdots+2020^3\right)-\left(1+2+3+\cdots2020\right)\)
\(=\left(1+2+\cdots+2020\right)^2-\left(1+2+3+\cdots+2020\right)\)
\(=\left(2020\cdot\frac{2021}{2}\right)^2-2020\cdot\frac{2021}{2}=\left(1010\cdot2021\right)^2-1010\cdot2021\)
XO
4 tháng 8 2020
Đặt A = 1.2 + 2.3 + 3.4 + ... + 2015.2016
=> 3A = 1.2.3 + 2.3.3 + 3.4.3 + ... + 2015.2016.3
=> 3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 2015.2016.(2017 - 2014)
=> 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 2015.2016.2017 - 2014.2015.2016
=> 3A = 2015.2016.2017
=> A = 2015.2017.672
=> A = 2 731 179 360
BM
1
AH
Akai Haruma
Giáo viên
4 tháng 10 2024
Lời giải:
$(1.2+2.3+3.4+...+2012.2013)-(2^2+3^2+...+2013^2)$
$=[(2-1).2+(3-1).3+(4-1).4+...+(2013-1).2013]-(2^2+3^2+...+2013^2)$
$=(2^2+3^2+4^2+...+2013^2)-(2+3+4+...+2013)-(2^2+3^2+...+2013^2)$
$=-(2+3+4+...+2013)$
$=1-(1+2+3+...+2013)$
$=1-2013.2014:2=1-2027091=-2027090$
PM
1.Thực hiện phép tính một cách hợp lý:
a,(1.2+2.3+3.4+...+2012.2013)-(\(^{2^2+3^2+4^2+...+2013^2}\))
0
ND
Nguyễn Đức Trí
VIP
31 tháng 7 2023
1)...=135+65+360+40+24=200+400+24=624
2)...=(8.125).(4.25)=1000.100=100000
4)...= (467+433)+(238+762)+3465=990+1000+3465=5455
5)...=129.(172-73+1)=129.100=12900
6)...=205-(1200-(16-6)3):40=205-(1200-1000):40=205-200:40=205-5=200
LV
1
MN
15 tháng 3 2016
(1.2+2.3+3.4+.....+2012.2013)-(22+32+42+......+20132)
= 1.2 + 2.3 + 3.4 +...+ 2012.2013 - 22 -32 - 42 -....-20132
=1.2 + 2.3 + 3.4 + ...+2012.2013 - 2.2 -3.3 - 4.4 -...- 2013.2013
=(1.2 - 2.2) + (2.3 - 3.3) + (3.4 - 4.4) + ...+(2012.2013 - 2013.2013)
=2.(1-2) + 3.(2-3) + 4.(3-4) +...+2013.(2012-2013)
=2.(-1) + 3.(-1) + 4.(-1) + ...+2013.(2012-2013)
= -2 - 3 - 4 -...- 2013
= -(2+3+4+...+2013)
= -[(2013+2).2012:2]
=-2027090
NK
1
NH
11 tháng 2 2020
\(1-2+3-4+...+199-200\)
\(=\left(1-2+3-4\right)+\left(5-6+7-8\right)+...+\left(197-198+199-200\right)\) ( 50 cặp )
\(=-2\times50\)
\(=-100\)
Sửa đề: \(A=1\cdot2+2\cdot3+\cdots+2012\cdot2013-\left(1^2+2^2+\cdots+2013^2\right)\)
Ta có: \(A=1\cdot2+2\cdot3+\cdots+2012\cdot2013-\left(1^2+2^2+\cdots+2013^2\right)\)
\(=1\left(1+1\right)+2\left(2+1\right)+\cdots+2012\left(2012+1\right)-\left(1^2+2^2+\cdots+2013^2\right)\)
\(=\left(1^2+2^2+\cdots+2012^2\right)+\left(1+2+\cdots+2012\right)-\left(1^2+2^2+\cdots+2013^2\right)\)
\(=\left(1+2+\cdots+2012\right)-2013^2=\frac{2012\cdot2013}{2}-2013^2\)
\(=1006\cdot2013-2013^2=2013\left(1006-2013\right)\)
\(=2013\cdot\left(-1007\right)=-2027091\)