Chứng tỏ \(A=5+5^2+5^3+...+5^{58}\)chia hết cho 30
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27 tháng 12 2017
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28 tháng 12 2017
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DN
1
30 tháng 10 2015
a) A=5(1+5)+53(1+5)+...+5199(1+5)
=(1+5)(5+53+....+5199) chia hết cho 6
b) A:31 dư 30 hay A-30 chia hết cho 31
Ta có A=5(1+5+52)+54(1+5+52)+57(1+5+52)+.....+598(1+5+52)
31(5+54+57+...+599) chia hết cho 31. Nên A chia cho 31 không dư
13 tháng 11 2018
A = 5 + 52 + 53 + ... + 512
A = (5 + 52) + (53 + 54) + ... + (511 + 512)
A = 30 + 52(5 + 52) + ... + 510(5 + 52)
A = 30 + 52.30 + ... + 510.30
A = 30(1 + 52 + ... + 510)
Vì 30(1 + 52 + ... + 510) chia hết cho 30 => A chia hết cho 30 (đpcm)
A = 5 + 52 + 53 + ... + 512
A = (5 + 52 + 53) + ... + (510 + 511 + 512)
A = 5(1 + 5 + 52) + ... + 510(1 + 5 + 52)
A = 5.31 + ... + 510.31
A = 31(5 + ... + 510)
Vì 31(5 + ... + 510) chia hết cho 31 => A chia hết cho 31 (đpcm)
13 tháng 11 2018
Ta có :
\(A=5+5^2+5^3+...+5^{12}\)
\(A=(5+5^2+5^3)+...+(5^{10}+5^{11}+5^{12})\)
\(A=5(1+5+5^2)+...+5^{10}(1+5+5^2)\)
\(A=5.31+...+5^{10}.31\)
\(A=(5+...+5^{10}).31\) chia hết cho 31
Ta có ;
\(A=5+5^2+5^3+...+5^{12}\)
\(A=5(1+5+5^2+...+5^{11})\) chia hết cho 5 ( 1 )
Ta lại có :
\(A=5+5^2+5^3+...+5^{12}\)
\(A=(5+5^2)+(5^3+5^4)+...+(5^{11}+5^{12})\)
\(A=5(1+5)+5^3(1+5)+...+5^{11}(1+5)\)
\(A=5.6+5^3.6+...+5^{11}.6\)
\(A=(5+5^3...+5^{11}).6\) chia hết cho 6 ( 2 )
Từ ( 1 ) và ( 2 ) ta có ;
\(A=5+5^2+5^3+...+5^{12}\) chia hết cho 5 và 6
=> \(A=5+5^2+5^3+...+5^{12}\)chia hết cho 30
TD
2
10 tháng 8 2016
\(A=5+5^2+5^3+5^4+..+5^7+5^8\)
\(A=\left(5+5^2\right)+5^2\left(5+5^2\right)+...+5^6\left(5^7+5^8\right)\)
\(A=30+5^2.30+...+5^6.30\)
\(A=30.\left(1+5^2+...+5^6\right)⋮30\)
\(\Rightarrow\) đpcm
NA
10 tháng 8 2016
\(A=5+5^2+5^3+...+5^8\\ A=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^7+5^8\right)\)
\(A=30+5^2\left(5+5^2\right)+5^4\left(5+5^2\right)+...+5^6\left(5+5^2\right)\\ A=30+5^2.30+5^4.30+...+5^6.30\)
\(A=30\left(1+5^2+5^4+...+5^6\right)\\ \Rightarrow A⋮30\)
17 tháng 8 2023
\(A=3+3^2+...+3^{101}+3^{102}\) (thêm 33 bi sót)
\(\Rightarrow A+1=1+3+3^2+...+3^{101}+3^{102}\)
\(\Rightarrow A+1=\dfrac{3^{102+1}-1}{3-1}\)
\(\Rightarrow A+1=\dfrac{3^{103}-1}{2}\)
\(\Rightarrow A=\dfrac{3^{103}-1}{2}-1\)
\(\Rightarrow A=\dfrac{3\left(3^{102}-1\right)}{2}\)
mà \(\left(3^{102}-1\right)\) không chia hết cho 2;4;5
\(\Rightarrow A=\dfrac{3\left(3^{102}-1\right)}{2}\) không chia hết cho 2;4;5
\(\Rightarrow A\) không chia hết cho 40 \(\left(vì40=2.4.5\right)\)
17 tháng 8 2023
\(B=4+4^2+4^3+...+4^{99}\)
\(\Rightarrow B=4\left(1+4^1+4^2\right)+4^4\left(1+4^1+4^2\right)...+4^{97}\left(1+4^1+4^2\right)\)
\(\Rightarrow B=4.21+4^4.21+...+4^{97}.21\)
\(\Rightarrow B=21\left(4+4^4+...+4^{97}\right)⋮21\)
\(\Rightarrow dpcm\)
HV
1
TS
10 tháng 4 2018
\(5+5^2+5^3+...5^{29}+5^{30}\)
\(=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{29}+5^{30}\right)\)
\(=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{29}\left(1+5\right)\)
\(=5.6+5^3.6+...+5^{29}.6⋮6\)
\(A=\left(5+5^2\right)+5^2\left(5+5^2\right)+.....+5^{56}\left(5+5^2\right)\)
\(\Rightarrow A=30\left(1+5^2+....+5^{56}\right)\) chia hết cho 30
=> A chia hết cho 30
(đpcm)
rõ hơn đi