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1) = 3n(32+1) - 2n(22+1)
2)A=m.n.p
\(\frac{m^2}{\frac{2^2}{5^2}}=\frac{n^2}{\frac{3^2}{4^2}}=\frac{p^2}{\frac{1^2}{6^2}}=\frac{m^2+n^2+p^2}{\frac{2^2}{5^2}+\frac{3^2}{4^2}+\frac{1^2}{6^2}}\)
3) \(\frac{a^2}{\text{\text{c}^2}}=\frac{\text{c}^2}{b^2}=\frac{a^2+\text{c}^2}{b^2+\text{c}^2}\)\(\frac{a^2}{\text{c}^2}=\frac{\text{c}^2}{b^2}=\frac{a^2+\text{c}^2}{\text{c}^2+b^2}\)
mà ab=c2
suy ra đpcm
a) \(3^{n+2}+3^n-2^{n+2}-2^n\)
\(=3^n\left(3^2+1\right)-2^n\left(2^2+1\right)\)
\(=3^n.10-2^n.5\)
\(=3^n.10-2^{n-1}.2.5\)chia hết cho 10
b)\(3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}\)
\(=3^{n+1}\left(3^2+1\right)+2^{n+2}\left(2+1\right)\)
\(=3^{n+1}.10+2^{n+2}.3\)
\(=3^n.3.2.5+2^{n+1}.2.3\)chia hết cho 6
\(A=3^{n+3}+2^{n+3}+3^{n+1}+2^{n+2}\)
\(A=3^n.3^3+2^n.2^3+3^n.3+2^n2^2\)
\(A=3^n.27+2^n.8+3^n.3+2^n.4\)
\(A=3^n.30+2^n.12\)
\(A=6\left(3^n.5+2^n.2\right)\)chia hết cho 6
1. Ta có:
\(\frac{a}{2}=\frac{b}{3}=\frac{c}{5}=\frac{2a}{4}=\frac{3b}{9}=\frac{5c}{25}\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta có:
\(\frac{2a}{4}=\frac{3b}{9}=\frac{5c}{25}=\frac{2a+3b-5c}{4+9-25}=\frac{-28}{-12}=\frac{7}{3}\)
\(\Rightarrow\frac{2a}{4}=\frac{7}{3}\Rightarrow2a=\frac{7}{3}.4=\frac{28}{3}\Rightarrow a=\frac{28}{3}:2=\frac{14}{3}\)
\(\Rightarrow\frac{3b}{9}=\frac{7}{3}\Rightarrow3b=\frac{7}{3}.9=21\Rightarrow b=21:3=7\)
\(\Rightarrow\frac{5c}{25}=\frac{7}{3}\Rightarrow5c=\frac{7}{3}.25=\frac{175}{3}\Rightarrow c=\frac{175}{3}:5=\frac{35}{3}\)
Vậy a = .......
b = ..........
c = ..............
Ta có:
\(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\frac{2a}{4}=\frac{3b}{9}=\frac{5c}{20}\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta có:
\(\frac{2a}{4}=\frac{3b}{9}=\frac{5c}{20}=\frac{2a+3b-5c}{4+9-20}=\frac{-28}{-7}=4\)
\(\Rightarrow\frac{2a}{4}=4\Rightarrow2a=4.4=16\Rightarrow a=16:2=8\)
\(\Rightarrow\frac{3b}{9}=4\Rightarrow3b=4.9=36\Rightarrow b=36:3=12\)
\(\Rightarrow\frac{5c}{20}=4\Rightarrow5c=4.20=80\Rightarrow c=80:5=16\)
Vậy a = 8
b = 12
c = 16
\(3xy-5=x^2+2y\Leftrightarrow xy-x^2+2xy-2y=5\Leftrightarrow x\left(y-x\right)+2y\left(x-y\right)=5\Leftrightarrow\left(2y-x\right)\left(x-y\right)=5\)
\(3^{n+2}-2^{n+2}+3^n-2^n=\left(3^{n+2}+3^n\right)-\left(2^{n+2}+2^n\right)=3^n\left(9+1\right)-2\left(2^{n+1}+2^{n-1}\right)\left(n\in Z^+\right)=3^n.10-2\left(4.2^{n-1}+2^{n-1}\right)=3^n.10-10.2^{n-1}=10\left(3^n-2^{n-1}\right)⋮10\)
b) 3n+2-2n+2+3n-2n = (3n+2+3n)+(-2n+2-2n) = (3n.32+3n)+[-2n.(-2)2-2n
= 3n (9+1) -2n(4+1)
=3n . 10 - 2n.5
= 3n.10 - 2n-1.10
= 10 ( 3n-2n-1) \(⋮\) 10
Vậy ...
3n + 3 + 3n + 1 + 2n + 3 + 2n + 2
= 3n.33 + 3n.3 + 2n.23 + 2n.22
= 3n.(27 + 3) + 2n.(8 + 4)
= 3n.30 + 2n.12
= 3n.5.6 + 2n.2.6
= 6.(3n.5 + 2n.2) \(⋮\) 6
\(A=1+3+3^2+3^3+...+3^{3n}+3^{3n+1}+3^{3n+2}\)
\(A=1.\left(1+3+9+\right)+3^3.\left(1+3+9\right)+3^6.\left(1+3+9\right)+...+3^{3n}.\left(1+3+9\right)\)
\(A=1.13+3^3.13+3^6.13+....+3^n.13\)
\(A=13.\left(1+3^3+3^6+...+3^{3n}\right)\)⋮ \(13\)
Vậy \(A\) ⋮ \(13\) ∀ \(n\)
A=3n+3 + 2n+3 +3n+1 +2n+2
=3n.33 +2n..23 + 3n .3 + 2n+22
= 3n(33+3) + 2n(23+22)
= 30.3n + 12.2n
=6 (5.3n + 2.2n) chia hết 6 Với moi n thuộc N (đpcm)