Monotonicity of a fraction.So I want to prove that the following fraction is monotone increasing, as a part of another proof, that's why I stumbled on: 4 n + 1 2 n + 1 ≥ 4 n 2 n I know it's basic, though how to prove it?

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Monotonicity of a fraction.So I want to prove that the following fraction is monotone increasing, as a part of another proof, that&#039;s why I stumbled on:            4              n        +        1                    2              n        +        1              ≥            4              n                    2              n            I know it&#039;s basic, though how to prove it?

Arcatuert3u · Accepted Answer

Both sides are positive, hence we may tyr to show the quotient is   ≥  1:            4              n        +        1                    2              n        +        1              &amp;gt;            4              n                    2              n                ⟺                      4                  n          +          1                            2                  n          +          1                                    4                  n                            2                  n                      =            4              n                    n      +      1        =                    16        n                    n        +        1              &amp;gt;  1The latter is certainly true for all   n  ≥  1  (because   16  n  &amp;gt;  n  +  1).

polivijuye · Accepted Answer

First, multiply both sides by   2  ×      4          −      n            n    +    1  . You are left with the following equivalent statement :  4  ≥      1    +          1      n      The function on the right hand side is strictly decreasing, and takes value        2   for   n  =  1, which is lower than 4. Therefore, the inequality holds for all integer   n  ≥  1

Monotonicity of a fraction. So I want to prove that the following fraction is monotone increasing,

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