Ta có:
\(P=\left(2+\sqrt{2}\right)^7+\left(2-\sqrt{2}\right)^7\)
\(P=2^7+7.2^6\sqrt{2}+21.2^5\left(\sqrt{2}\right)^2+...+7.2\left(\sqrt{2}\right)^6+\left(\sqrt{2}\right)^7\)\(+2^7-7.2^6\sqrt{2}+21.2^5\left(\sqrt{2}\right)^2-...+7.2\left(\sqrt{2}\right)^6-\left(\sqrt{2}\right)^7\)
\(P=2.2^7+2.21.2^5.\left(\sqrt{2}\right)^2+2.35.2^3.\left(\sqrt{2}\right)^4+2.7.2.\left(\sqrt{2}\right)^6\)
\(P=2^8+21.2^7+35.2^6+7.2^5\)
\(P=5408\)
\(\Rightarrow\left(2+\sqrt{2}\right)^7=5408-\left(2-\sqrt{2}\right)^7\)
Do \(0< \left(2-\sqrt{2}\right)^7< 1\) nên suy ra \(5047< \left(2+\sqrt{2}\right)^7< 5048\)
Vậy số nguyên lớn nhất không vượt quá \(\left(2+\sqrt{2}\right)^7\) là 5047.
(Sau này ta kí hiệu như thế này cho gọn.)
