Whether the sequence -1,1,3,5,.... is arithmetic, geometric or neither

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Lilia Horton · Accepted Answer

A sequence             a      n       is:Arithmetic if the difference between two consecute elements, i.e.             a      n        -          a              n        -        1            , is constant.Geometric if the ratio between two consecute elements, i.e.                     a        n                    a                  n          -          1                    , is constant.Let&#039;s test both to get the answer.Difference testLet&#039;s consider the difference between every element and the following one:            a      2        -          a      1        =    1    -          (      -      1      )        =    1    +    1    =    2              a      3        -          a      2        =    3    -    1    =    2              a      4        -          a      3        =    5    -    3    =    2  The difference remains constant (it&#039;s always 2), so we may already conclude that the sequence is arithmetic. Let&#039;s do the ratio test nevertheless:Ratio testLet&#039;s consider the ratio between every element and the following one:                    a        2                    a        1              =          1              -        1              =    -    1                      a        3                    a        2              =          3      1        =    3                      a        4                    a        3              =          5      3        =    1.666    ...  The ratio is always different, so the sequence is not geometric, as expected.

Whether the sequence -1,1,3,5,.... is arithmetic, geometric or neither

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