(0,5 điểm) Cho S = 1 4 + 2 4 2 + 3 4 3 + 4 4 4 + . . . . + 2023 4 2023 . Chứng minh S < 1 2 .

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(0,5 điểm) Cho \(S = \frac{1}{4} + \frac{2}{{{4^2}}} + \frac{3}{{{4^3}}} + \frac{4}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2023}}}}\). Chứng minh \(S < \frac{1}{2}.\)

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Hướng dẫn giải

Ta có: \(S = \frac{1}{4} + \frac{2}{{{4^2}}} + \frac{3}{{{4^3}}} + \frac{4}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2023}}}}\)

\(4S = 4\left( {\frac{1}{4} + \frac{2}{{{4^2}}} + \frac{3}{{{4^3}}} + \frac{4}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2023}}}}} \right)\)

\(4S = 1 + \frac{2}{4} + \frac{3}{{{4^2}}} + \frac{4}{{{4^3}}} + \frac{5}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2022}}}}\)

\(4S - S = 1 + \frac{2}{4} + \frac{3}{{{4^2}}} + \frac{4}{{{4^3}}} + \frac{5}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2022}}}} - \left( {\frac{1}{4} + \frac{2}{{{4^2}}} + \frac{3}{{{4^3}}} + \frac{4}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2023}}}}} \right)\)

\(3S = 1 + \frac{2}{4} + \frac{3}{{{4^2}}} + \frac{4}{{{4^3}}} + \frac{5}{{{4^4}}} + .... + \frac{{2023}}{{{4^{2022}}}} - \frac{1}{4} - \frac{2}{{{4^2}}} - \frac{3}{{{4^3}}} - \frac{4}{{{4^4}}} - .... - \frac{{2023}}{{{4^{2023}}}}\)

\(3S = 1 + \left( {\frac{2}{4} - \frac{1}{4}} \right) + \left( {\frac{3}{{{4^2}}} - \frac{2}{{{4^2}}}} \right) + \left( {\frac{4}{{{4^3}}} - \frac{3}{{{4^3}}}} \right) + \left( {\frac{5}{{{4^4}}} - \frac{4}{{{4^4}}}} \right) + .... + \left( {\frac{{2023}}{{{4^{2022}}}} - \frac{{2022}}{{{4^{2022}}}}} \right) - \frac{{2023}}{{{4^{2023}}}}\)

\(3S = 1 + \frac{1}{4} + \frac{1}{{{4^2}}} + \frac{1}{{{4^3}}} + \frac{1}{{{4^4}}} + .... + \frac{1}{{{4^{2022}}}} - \frac{{2023}}{{{4^{2023}}}}\)

Nhận thấy \(3S < 1\).

Đặt \(A = 1 + \frac{1}{4} + \frac{1}{{{4^2}}} + \frac{1}{{{4^3}}} + \frac{1}{{{4^4}}} + .... + \frac{1}{{{4^{2022}}}}\)

\(4A = 4 + 1 + \frac{1}{4} + \frac{1}{{{4^2}}} + \frac{1}{{{4^3}}} + \frac{1}{{{4^4}}} + .... + \frac{1}{{{4^{2021}}}}\)

\(4A - A = 4 + 1 + \frac{1}{4} + \frac{1}{{{4^2}}} + \frac{1}{{{4^3}}} + \frac{1}{{{4^4}}} + .... + \frac{1}{{{4^{2021}}}} - \left( {1 + \frac{1}{4} + \frac{1}{{{4^2}}} + \frac{1}{{{4^3}}} + \frac{1}{{{4^4}}} + .... + \frac{1}{{{4^{2022}}}}} \right)\)

\(3A = 4 - \frac{1}{{{4^{2022}}}}\)

Nhận thấy \(4 - \frac{1}{{{4^{2022}}}} < 4\) hay \(3A < 4\) suy ra \(A < \frac{4}{3}\).

Do đó, \(3S < A\) nên \(S < \frac{A}{3}\) hay \(S < \frac{4}{9} < \frac{4}{8} = \frac{1}{2}.\)

Vậy \(S < \frac{1}{2}.\)