What is the least positive integer n for which ( n + 1 ) 1 3 - n 1 3 &lt; 1 12 satisfies?

Question

What is the least positive integer n for which                     (        n        +        1        )                              1          3                      -          n                        1          3                      &amp;lt;          1      12       satisfies?

Cristian Hamilton · Accepted Answer

Step 1Note that:                    (        x        +                  1          12                )            3        =          x      3        +    3          (              1        12            )              x      2        +    3                  (                  1          12                )            2        x    +                  (                  1          12                )            3                                                (            x            +                          1              12                        )                    3                      =          x      3        +          1      4              x      2        +          1      48        x    +          1      1728      Given:                    (        n        +        1        )                              1          3                      -          n                        1          3                      &amp;lt;          1      12      Add             n                        1          3                     to both sides to get:                    (        n        +        1        )                              1          3                      &amp;lt;          n                        1          3                      +          1      12      Cube both sides to get:                                                                        n                                                          +    1    &amp;lt;                                                                      n                                                          +          1      4              n                        2          3                      +          1      48              n                        1          3                      +          1      1728      Subtract n from both sides and multiply both sides by 1728 to get:      1728    &amp;lt;    432          n                        2          3                      +    36          n                        1          3                      +    1  Subtract 1728 from both sides, transpose and substitute      x    =          n                        1          3                    to get:      432          x      2        +    36    x    -    1727    &amp;gt;    0  Rather than solve this properly, let&#039;s approximate the positive zero:      x    ≈                            1727          432                      ≈                  4              =    2  Then       n    =          x      3        ≈    8  We find:                    (        8        +        1        )                              1          3                      -          8                        1          3                      =                  9                    3              -    2    ≈    0.08008    &amp;lt;    0.08                  3            ¯        =          1      12

Kyla Ayers · Accepted Answer

Step 1Let   n  &amp;gt;  1 . Then       1    n    &amp;lt;  1 .Use                     (        1        +        x        )            m        =    1    +    m          C      1        x    +    m          C      2              x      2        +    ...    ,    x    ∈          (      -      1      ,      1      ]                          (        n        +        1        )                              1          3                          =          n                        1          3                                    (        1        +                  1          n                )                    -                  2          3                          =          n                        1          3                            (      1      +              (                  1          3                )                    (                  1          n                )            +                                    (                          1              3                        )                                (                          1              3                        -            1            )                                    2          !                            (                  1                      n            2                          )            +      ...      .      .            =          n                        2          3                      +          (              1        3            )              n              -                  2          3                      -          (              1        9            )              n              -                  5          3                      +    ...  Here, the absolute values of the terms form a diminishing sequence. It follows that                    (        n        +        1        )                              1          3                      -          n                        1          3                      &amp;lt;          (              1        3            )              n              -                  2          3                      &amp;lt;          1      12      And so,             n              -                  2          3                      &amp;lt;          1      4        →          n                        2          3                      &amp;gt;    4    →    n    &amp;gt;          4                        3          2                      =    8  As the value of the expression, for   n  =  7 ,      =    2    -          7                        1          3                      =    0.087    ...    &amp;gt;          1      12        =    0.0833    .    .  the answer is 8.

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