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Ta có : P = 1 + 2 + 22 + 23 + ....... + 298 + 2100
=> P = (1 + 2) + (22 + 23) + ....... + (298 + 2100)
=> P = (1 + 2) + 22.(1 + 2) + ..... + 298(1 + 2)
=> P = 3 + 22.3 + ..... + 298.3
=> P = 3(1 + 22 + ..... + 298) chia hết cho 3
Sn = 1 + a + a2 + a3 + .................. + an
=> 2.Sn = a + a2 + a3 + .................... + an + 1
=> 2.Sn - Sn = an + 1 - 1
=> Sn = an + 1 - 1
a)Ta có: \(5^{36}=5^{3.12}=\left(5^3\right)^{12}=125^{12}\)
\(11^{24}=11^{2.12}=\left(11^2\right)^{12}=121^{12}\)
Vì \(125>121\Rightarrow125^{12}>121^{12}\)
\(\Rightarrow5^{36}>11^{24}\)
b) Ta có: \(625^5=\left(5^4\right)^5=5^{20}\)
\(125^7=\left(5^3\right)^7=5^{21}\)
Vì \(20< 21\Rightarrow5^{20}< 5^{21}\)
\(\Rightarrow625^5< 125^7\)
c) Ta có: \(3^{2n}=\left(3^2\right)^n=9^n\)
\(2^{3n}=\left(2^3\right)^n=8^n\)
Vì \(9>8\Rightarrow9^n>8^n\)( do \(n>0\))
\(\Rightarrow3^{2n}>2^{3n}\)
d)Ta có: \(5^{23}=5.5^{22}< 6.5^{22}\)
\(\Rightarrow5^{23}< 6.5^{22}\)
a. 5^36=(5^3)^12
=125^12
11^24=(11^2)^12
= 121^12
Vì 125^12>121^12 nên 5^36>11^24
b. Ta có: 625^5 =(5^4)^5
= 5^20
125^7=(5^3)^7
= 5^21
Vì 5^20<5^21 nên 625^5<125^7
\(S=1^2+2^2+3^2+...+51^2\)
\(S=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+51\left(52-1\right)\)
\(S=1.2-1.1+2.3-1.2+3.4-1.3+...+51.52-1.51\)
\(S=\left(1.2+2.3+3.4+...+51.52\right)-\left(1+2+3+...+51\right)\)
Đặt \(A=1.2+2.3+3.4+...+51.52\)
\(3A=1.2.3+2.3.3+3.4.3+...+51.52.3\)
\(3A=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+51.52.\left(53-50\right)\)
\(3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+51.52.53-50.51.52\)
\(3A=51.52.53\)
\(A=\frac{51.52.53}{3}=46852\)
\(\Rightarrow\)\(S=A-\left(1+2+3+...+51\right)=46852-1326=45526\)
Vậy \(S=45526\)
Chúc bạn học tốt ~
\(S=1.1+2.2+3.3+.......+51.51\)
\(S=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+.............+51\left(52-1\right)\)
\(S=1.2+2.3+3.4+.....+51.52-1-2-3-4-5-........-51\)
\(3S=1.2.\left(3-0\right)+.........+51.52.\left(53-50\right)-\left(1+2+3+.......+51\right).3\)
\(3S=51.52.53-52.51:2.3\)
\(...............................................\)