Hướng dẫn giải Bài 4 (Trang 136 SGK Toán Giải tích 12)

Bài 4 (Trang 136 SGK Toán Giải tích 12): Tính <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mn>3</mn></msup><mo>,</mo><mo> </mo><msup><mi>i</mi><mn>4</mn></msup><mo>,</mo><mo> </mo><msup><mi>i</mi><mn>5</mn></msup></math>. Nếu cách tính <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mi>n</mi></msup></math>với n là một số tự nhiên tuỳ ý.  Hướng dẫn giải: Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">i</mi><mn>3</mn></msup><mo>=</mo><msup><mi mathvariant="normal">i</mi><mn>2</mn></msup><mo>.</mo><mi mathvariant="normal">i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">i</mi><mspace linebreak="newline"/><msup><mi mathvariant="normal">i</mi><mn>4</mn></msup><mo>=</mo><msup><mrow><mo>(</mo><msup><mi mathvariant="normal">i</mi><mn>2</mn></msup><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn><mspace linebreak="newline"/><msup><mi mathvariant="normal">i</mi><mn>5</mn></msup><mo>=</mo><msup><mi mathvariant="normal">i</mi><mn>4</mn></msup><mo>.</mo><mi mathvariant="normal">i</mi></math> Lấy n chia cho 4: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">n</mi><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><mi mathvariant="normal">k</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">r</mi><mo>,</mo><mo> </mo><mn>0</mn><mo> </mo><mo>≤</mo><mo> </mo><mi mathvariant="normal">r</mi><mo> </mo><mo>≤</mo><mo> </mo><mn>3</mn></math>                             <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mi mathvariant="normal">i</mi><mrow><mn>4</mn><mi mathvariant="normal">k</mi><mo>+</mo><mn>5</mn></mrow></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mi mathvariant="normal">i</mi><mi mathvariant="normal">r</mi></msup><msup><mfenced><msup><mi mathvariant="normal">i</mi><mn>4</mn></msup></mfenced><mi mathvariant="normal">k</mi></msup><mo> </mo><mo>=</mo><mo> </mo><msup><mi mathvariant="normal">i</mi><mi mathvariant="normal">r</mi></msup></math> <ul> <li>Nếu r=0 thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math></li> <li>Nếu r=1 thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mi>n</mi></msup><mo>=</mo><mi>i</mi></math></li> <li>Nếu r=2 thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mi>n</mi></msup><mo>=</mo><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mo>-</mo><mn>1</mn></math></li> <li>Nếu r=3 thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><msup><mi>i</mi><mn>3</mn></msup><mo>=</mo><mo>-</mo><mi>i</mi></math></li> </ul>