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\(3.3^{n-1}\left(6.3^{n+2}+3\right)-2.3^n\left(3^{n+3}-1\right)=405\)
\(\Leftrightarrow18.3^{2n+1}+3.3^n-2.3^{2n+3}+2.3^n=405\)
\(\Leftrightarrow54.3^{2n}+5.3^n-2.3^3.3^{2n}=405\)
\(\Leftrightarrow3^n=81\)
\(\Leftrightarrow n=4\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(3.3^{n-1}\left(6.3^{n+2}+3\right)-2.3^n\left(3^{n+3}-1\right)=405\)
\(\Leftrightarrow18.3^{2n+1}+3.3^n-2.3^{2n+3}+2.3^n=405\)
\(\Leftrightarrow54.3^{2n}+5.3^n-2.3^3.3^{2n}=405\)
\(\Leftrightarrow3^n=81\)
\(\Leftrightarrow n=4\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Lời giải:
Đặt $3^n=a(a>0)$ thì pt trở thành:
$a^2+a+1=757$
$\Leftrightarrow a^2+a-756=0$
$\Leftrightarrow (a-27)(a+28)=0$
Vì $a>0$ nên $a+28\neq 0$
$\Rightarrow a-27=0\Rightarrow a=27$
$\Leftrightarrow 3^n=27=3^3\Rightarrow n=3$
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: n + 2 2 – (n – 3)(n + 3) ≤ 40
⇔ n 2 + 4n + 4 – n 2 + 9 ≤ 40
⇔ 4n < 40 – 13
⇔ n < 27/4
Vậy các số tự nhiên cần tìm là 0; 1; 2; 3; 4; 5; 6.
![](https://rs.olm.vn/images/avt/0.png?1311)
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Ta có
\(a+b+c=1\)
\(\Rightarrow\left(a+b+c\right)^3=a^3+b^3+c^3+3\left(a+b\right)\left(b+c\right)\left(a+c\right)=1\)
Mà \(a^3+b^3+c^3=1\)
\(\Rightarrow3\left(a+b\right)\left(b+c\right)\left(a+c\right)=0\)
\(\Leftrightarrow\left(a+b\right)\left(b+c\right)\left(a+c\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=-b\\b=-c\\c=-a\end{matrix}\right.\)
Do a;b ;c bình đẳng nên giả sử a = - b
\(\Rightarrow a+b+c=1\)
\(\Leftrightarrow-b+b+c=1\Leftrightarrow c=1\)
\(A=a^n+b^n+c^n\) Do n là số TN lẻ nên
\(A=a^n+b^n+c^n=\left(-b\right)^n+b^n+c^n=-b^n+b^n+c^n=c^n=1^n=1\)
![](https://rs.olm.vn/images/avt/0.png?1311)
ĐK \(n\ge0\)
Ta có \(3.3^{n-1}\left(6.3^{n+2}+3\right)-2.3^n\left(3^{n+3}-1\right)=405\)
\(\Leftrightarrow3^n\left(6.9.3^n+3\right)-2.3^n\left(27.3^n-1\right)=405\)
\(\Leftrightarrow54.3^{2n}+3.3^n-54.3^{2n}+2.3^n=405\Leftrightarrow5.3^n=405\)
\(\Leftrightarrow3^n=81=3^4\Leftrightarrow n=4\left(tm\right)\)
Vậy \(n=4\)
3n + 3n + 3n = 313
3.3n = 313
3n + 1 = 313
=> n + 1 = 13
=> n = 12
3n + 3n + 3n = 313
3n x 3 = 1594323
3n = 1594323 : 3
3n = 531441
3n = 312
Vậy n = 12