Hướng dẫn giải Bài 6 (Trang 58 SGK Toán Đại số & Giải tích 11)

Chứng minh rằng: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><msup><mn>11</mn><mn>10</mn></msup><mo>-</mo><mn>1</mn><mo> </mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>a</mi><mo> </mo><mi>h</mi><mi>ế</mi><mi>t</mi><mo> </mo><mn>100</mn><mo> </mo><mspace linebreak="newline"/><mi>b</mi><mo>,</mo><mo> </mo><msup><mn>101</mn><mn>100</mn></msup><mo>-</mo><mn>1</mn><mo> </mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>a</mi><mo> </mo><mi>h</mi><mi>ế</mi><mi>t</mi><mo> </mo><mn>10</mn><mo> </mo><mn>000</mn><mo> </mo><mspace linebreak="newline"/><mi>c</mi><mo>,</mo><mo> </mo><msqrt><mn>10</mn></msqrt><mo> </mo><mfenced open="[" close="]"><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup></mrow></mfenced><mo> </mo><mi>l</mi><mi>à</mi><mo> </mo><mi>m</mi><mi>ộ</mi><mi>t</mi><mo> </mo><mi>s</mi><mi>ố</mi><mo> </mo><mi>n</mi><mi>g</mi><mi>u</mi><mi>y</mi><mi>ê</mi><mi>n</mi><mo> </mo></math> Giải  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>T</mi><mi>a</mi><mo> </mo><mi>c</mi><mi>ó</mi><mo> </mo><mo>:</mo><mo> </mo><msup><mn>11</mn><mn>10</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><msup><mfenced><mrow><mn>10</mn><mo>+</mo><mn>1</mn></mrow></mfenced><mn>10</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mn>10</mn></munderover><msubsup><mo largeop="true">C</mo><mn>10</mn><mi>k</mi></msubsup><mo>-</mo><mn>1</mn><mspace linebreak="newline"/><mo>=</mo><mo> </mo><mn>10</mn><mo>.</mo><msubsup><mo largeop="true">C</mo><mn>10</mn><mn>1</mn></msubsup><mo>-</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><mn>10</mn></munderover><msubsup><mo largeop="true">C</mo><mn>10</mn><mi>k</mi></msubsup><msup><mn>10</mn><mi>k</mi></msup><mo> </mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>a</mi><mo> </mo><mi>h</mi><mi>ế</mi><mi>t</mi><mo> </mo><mi>c</mi><mi>h</mi><mi>o</mi><mo> </mo><mn>100</mn><mo> </mo><mspace linebreak="newline"/><mi>V</mi><mi>ì</mi><mo> </mo><mn>10</mn><msubsup><mo largeop="true">C</mo><mn>10</mn><mn>1</mn></msubsup><mo>=</mo><mn>100</mn><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><mn>10</mn></munderover><msubsup><mo largeop="true">C</mo><mn>10</mn><mi>k</mi></msubsup><msup><mn>10</mn><mi>k</mi></msup><mo>⋮</mo><mn>100</mn><mo> </mo></math> b, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>a</mi><mo> </mo><mi>c</mi><mi>ó</mi><mo> </mo><msup><mn>100</mn><mn>100</mn></msup><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mo>=</mo><msup><mfenced><mrow><mn>100</mn><mo>+</mo><mn>1</mn></mrow></mfenced><mn>100</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mi>k</mi></msubsup><msup><mn>100</mn><mi>k</mi></msup><mo> </mo><mo>-</mo><mn>1</mn><mspace linebreak="newline"/><mo>=</mo><mn>1</mn><mo>+</mo><msubsup><mo largeop="true">C</mo><mn>100</mn><mn>1</mn></msubsup><mo>.</mo><mn>100</mn><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mi>k</mi></msubsup><msup><mn>100</mn><mi>k</mi></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>1000</mn><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mi>k</mi></msubsup><msup><mn>100</mn><mi>k</mi></msup><mo>⋮</mo><mn>10000</mn><mspace linebreak="newline"/></math> c,  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>a</mi><mo> </mo><mi>c</mi><mi>ó</mi><mo> </mo><mspace linebreak="newline"/><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mi>k</mi></msubsup><msup><mfenced><msqrt><mn>10</mn></msqrt></mfenced><mi>k</mi></msup><mspace linebreak="newline"/><mo>=</mo><munderover><mo>∑</mo><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>=</mo><mn>2</mn><mi>l</mi><mo>≤</mo><mn>100</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi></mrow></msubsup><mo>.</mo><mn>10</mn><mi>l</mi><mo>+</mo><munderover><mo>∑</mo><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>=</mo><mn>2</mn><mi>l</mi><mo>≤</mo><mn>100</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>.</mo><msup><mn>10</mn><mi>l</mi></msup><mo>.</mo><msqrt><mn>10</mn></msqrt><mspace linebreak="newline"/><mo>=</mo><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>0</mn></mrow><mn>50</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi></mrow></msubsup><mo>.</mo><msup><mn>10</mn><mi>l</mi></msup><mo>+</mo><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>0</mn></mrow><mn>49</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>.</mo><msup><mn>10</mn><mi>l</mi></msup><mo>.</mo><msqrt><mn>10</mn></msqrt><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>1</mn></mfenced><mspace linebreak="newline"/><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mn>100</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mi>k</mi></msubsup><msup><mfenced><mrow><mo>-</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mi>k</mi></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mn>50</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi></mrow></msubsup><mo>.</mo><msup><mn>10</mn><mi>l</mi></msup><mo>-</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mn>49</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>.</mo><msup><mn>10</mn><mi>l</mi></msup><mo>.</mo><msqrt><mn>10</mn></msqrt><mo> </mo><mo> </mo><mfenced><mn>2</mn></mfenced><mspace linebreak="newline"/><mi>L</mi><mi>ấ</mi><mi>y</mi><mo> </mo><mfenced><mn>1</mn></mfenced><mo> </mo><mi>t</mi><mi>r</mi><mi>ừ</mi><mo> </mo><mfenced><mn>2</mn></mfenced><mo> </mo><mi>t</mi><mi>a</mi><mo> </mo><mi>đ</mi><mi>ư</mi><mi>ợ</mi><mi>c</mi><mo> </mo><mspace linebreak="newline"/><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup><mo>=</mo><mn>2</mn><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>0</mn></mrow><mn>49</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>.</mo><msup><mn>10</mn><mi>l</mi></msup><mo>.</mo><msqrt><mn>10</mn></msqrt><mspace linebreak="newline"/><mi>S</mi><mi>u</mi><mi>y</mi><mo> </mo><mi>r</mi><mi>a</mi><mo> </mo><msqrt><mn>10</mn></msqrt><mo> </mo><mfenced open="[" close="]"><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>10</mn></msqrt></mrow></mfenced><mn>100</mn></msup></mrow></mfenced><mo>=</mo><mn>2</mn><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>0</mn></mrow><mn>49</mn></munderover><msubsup><mo largeop="true">C</mo><mn>100</mn><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mn>10</mn><mrow><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msup><mo> </mo><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>