Given the sequence { 6 , 18 , 54 , 162 , ... } , find the next three terms and an expression for the n t h term?

Question

Given the sequence             {      6      ,      18      ,      54      ,      162      ,      ...      }      , find the next three terms and an expression for the             n              t        h             term?

Kathleen Garza · Accepted Answer

The sequence:            {      6      ,      18      ,      54      ,      162      ,      ...      }      clearly is not linear or quadratic as the terms increase too rapidly, we note that they are all even and factors of       6  , so we let us see how factoring out       6   helps to establish the pattern:            {      6      ,      6      ×      3      ,      6      ×      9      ,      6      ×      27      ,      ...      }      and we can now see that the factor of       6   is multiplied by a power of       3  , giving us:            {      6      ,      6      ×              3        1            ,      6      ×              3        2            ,      6      ×              3        3            ,      ...      }      And we also now that             3      0        =    1  , so we can modify the first term as follows:            {      6      ×              3        0            ,      6      ×              3        1            ,      6      ×              3        2            ,      6      ×              3        3            ,      ...      }      And then we can see that the pattern is established and that the             n              t        h             term is given by (assuming we start the sequence at       n    =    1  :            u      n        =    6    ×          3              n        -        1            In order to create the following three terms, first let&#039;s make sure that this works:      n    =    1    ⇒          u      1        =    6    ×          3      0        =    6        n    =    2    ⇒          u      2        =    6    ×          3      1        =    18        n    =    3    ⇒          u      3        =    6    ×          3      2        =    54        n    =    4    ⇒          u      4        =    6    ×          3      3        =    162        n    =    5    ⇒          u      5        =    6    ×          3      4        =    486        n    =    6    ⇒          u      6        =    6    ×          3      5        =    1458        n    =    7    ⇒          u      7        =    6    ×          3      6        =    4374

Destiney Rivers · Accepted Answer

nice

Given the sequence { 6,18,54,162,... } , find the next three terms and an expression for the n^(th) term?

Answered question

Answer & Explanation

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