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2.
a,\(50-\left[\left(50-2^3.5\right):2+3\right]\)
\(=50-\left[\left(50-40\right):2+3\right]\)
\(=50-\left(10:2+3\right)\)
\(=50-8\)
\(=42\)
b,\(8697-\left[3^7:3^5+2\left(13-3\right)\right]\)
\(=8697-\left(3^2+2.10\right)\)
\(=8697-\left(9+20\right)\)
\(=8697-29\)
\(=8668\)
c,\(205-\left[1200-\left(4^2-2.3\right)^3\right]:40\)
\(=205-200:40\)
\(=200\)
2)
a) \(50-\left[\left(50-2^3.5\right):2+3\right]\)
\(=50-\left[\left(50-8.5\right):2+3\right]\)
\(=50-\left[\left(50-40\right):2+3\right]\)
\(=50-\left(10:2+3\right)\)
\(=50-\left(5+3\right)\)
\(=50-8\)
\(=42\)
b) \(8697-\left[3^7:3^5+2\left(13-3\right)\right]\)
\(=8697-\left(3^7:3^5+2.10\right)\)
\(=8697-\left(3^{7-5}+2.10\right)\)
\(=8697-\left(3^2+2.10\right)\)
\(=8697-\left(9+2.10\right)\)
\(=8697-\left(9+20\right)\)
\(=8697-29\)
\(=8668\)
c) \(205-\left[1200-\left(4^2-2.3\right)^3\right]:40\)
\(=205-\left[1200-\left(16-2.3\right)^3\right]:40\)
\(=205-\left[1200-\left(16-6\right)^3\right]:40\)
\(=205-\left(1200-10^3\right):40\)
\(=205-\left(1200-1000\right):40\)
\(=205-200:40\)
\(=205-5\)
\(=200\)
\(B=2+2^2+2^3+...+2^{60}\)
\(=2\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)
\(=7\cdot\left(2+...+2^{58}\right)⋮7\)
1)
a)\(B=3+3^3+3^5+3^7+.....+3^{1991}\)
\(\Leftrightarrow B=3\left(1+3^2+3^4+3^6+.....+3^{1990}\right)\)
Vì \(3\left(1+3^2+3^4+3^6+.....+3^{1990}\right)\)chia hết cho 3 nên \(B⋮3\)
\(B=3+3^3+3^5+3^7+.....+3^{1991}\)
\(\Leftrightarrow B=\left(3+3^3+3^5+3^7\right)+.....+\left(3^{1988}+3^{1989}+3^{1990}+3^{1991}\right)\)
\(\Leftrightarrow B=3\left(1+3^2+3^4+3^6\right)+.....+3^{1988}\left(1+3^2+3^4+3^6\right)\)
\(\Leftrightarrow B=3.820+.....+3^{1988}.820\)
\(\Leftrightarrow B=3.20.41+.....+3^{1988}.20.41\)
Vì \(3.20.41+.....+3^{1988}.20.41\) chia hết cho 41 nên \(B⋮41\)
a, 11 + 112 + 113 + ... + 117 + 118
= (11 + 112) + (113 + 114) + ... + (117 + 118)
= 11(1 + 11) + 113(1 + 11) + ... + 117(1 + 11)
= 11.12 + 113.12 + .... + 117.12
= 12(11 + 113 + ... + 117) chia hết cho 12
b, 7 + 72 + 73 + 74
= (7 + 73) + (72 + 74)
= 7(1 + 72) + 72(1 + 72)
= 7.50 + 72.50
= 50(7 + 72) chia hết cho 50
c, 3 + 32 + 33 + 34 + 35 + 36
= (3 + 32 + 33) + (34 + 35 + 36)
= 3(1 + 3 + 32) + 34(1 + 3 + 32)
= 3.13 + 34.13
= 13(3 + 34) chia hết cho 13
Vì 13 là lẻ \(\Rightarrow\) 13, 132, 133, 134, 135, 136 là lẻ.
Mà lẻ + lẻ + lẻ + lẻ + lẻ + lẻ = chẵn nên 13 + 132 + 133 + 134 + 135 + 136 là chẵn. \(\Rightarrow\) 13 + 132 + 133 + 134 + 135 + 136 \(⋮\) 2
\(\Rightarrow\) ĐPCM
