a) \(A=2^{n-1}+2.2^{n+3}-8.2^{n-4}-16.2^n\)

\(=2^{n-1}+2^{n+3+1}-2^{n-4+3}-2^{n+4}\)

\(=2^{n-1}+2^{n+4}-2^{n-1}-2^{n+4}\)

\(=0\)

b) \(B=\left(3^{n+1}-2.2^n\right)\left(3^{n+1}+2.2^n\right)-3^{2n+2}+\left(8.2^{n-2}\right)^2\)

\(=\left(3^{n+1}-2^{n+1}\right)\left(3^{n+1}-2^{n+1}\right)-3^{2n+2}+2^{2n+2}\)

\(=3^{2n+2}-2^{2n+2}-3^{2n+2}+2^{2n+2}\)

\(=0\)

a,

\(A=2^{n-1}+2.2^{n+3}-8.2^{n-4}-16.2^n\)

\(=2^{n-1}+2^{n+3+1}-2^{n-4+3}-2^{n+4}\)

\(=2.2^{n-1}+2.2^{n+4}=2^n+2^{n+5}\)

b,

\(B=\left(3^{n+1}-2.2^n\right)\left(3^{n+1}+2.2^n\right)-3^{2n+2}+\left(8.2^{n-2}\right)^2\)

\(=\left(3^{n+1}\right)^2-\left(2.2^n\right)^2-\left(3^{n+1}\right)^2+\left(2^{n-2+3}\right)^2\)

\(=-2^{n+1}+2^{n+1}=0\)

\(\left(3^{n+1}-2.2^n\right)\left(3.3^n+2^{n+1}\right).3^{2n+2}+\left(8.2^{n-2}.3^{n+1}\right)^2\)

\(=\left(3^{n+1}-2^{n+1}\right)\left(3^{n+1}+2^{n+1}\right).3^{2n+2}+\left(2^{n+1}.3^{n+1}\right)^2\)

\(=\left(3^{2n+2}-2^{2n+2}\right).3^{2n+2}+2^{2n+2}.3^{2n+2}\)

\(=3^{2\left(2n+2\right)}-2^{2n+2}.3^{2n+2}+2^{2n+2}.3^{2n+2}\)

\(=3^{2\left(2n+2\right)}=\left(3^{2n+2}\right)^2\).

Ta thấy \(\left(3^{2n+2}\right)^2\)luôn là 1 số chính phương với mọi n\(\in\)N

Nên ta có ĐPCM.

Lời giải:
Đặt biểu thức đã cho là $A$

Ta viết lại biểu thức thành:

\(A=(3^{n+1}-2^{n+1})(3^{n+1}+2^{n+1}).3^{2(n+1)}+(2^{n+1}.3^{n+1})^2\)

Đặt \(3^{n+1}=a; 2^{n+1}=b\Rightarrow A=(a-b)(a+b)a^{2}+(ba)^2\)

\(=(a^2-b^2)a^2+a^2b^2=a^4=(a^2)^2\)

Do đó biểu thức đã cho là một số chính phương.

Ta có đpcm.

= 0 nha bạn