Write the rule for the nth term given #2,-4,8,-10,14,...

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yamalwg · Accepted Answer

Given:2,−4,8,−10,14Look at the sequence of differences between successive terms:−6,12,−18,24The absolute values of these are in arithmetic progression with common difference 6, but with alternating signs.Hence we can express our original sequence by a recursive rule:            {                                                  a              1                        =            2                                                              a                              n                +                1                                      =                          a              n                        +                                          (                -                1                )                            n                        6            n                              How about a general formula for             a      n       in terms of n?Looking at the alternate terms we can observe that we have two interwoven arithmetic sequences - one (2,8,14,...) with a common difference of 6 and the other (−4,−10,−16,...) with a common difference of −6.Hence we can write:            a      n        =          {                                    6            n            -            4                                         if                                       n                                         is odd                                                            8            -            6            n                                         if                                       n                                         is even                                          Note that if we subtract 2 from each of the expressions for odd and even n then we can make the expressions look a little more similar:            a      n        =          {                                    2            -                          (              6              -              6              n              )                                                     if                                       n                                         is odd                                                            2            +                          (              6              -              6              n              )                                                     if                                       n                                         is even                                          and hence:            a      n        =    2    +                  (        -        1        )            n        6          (      1      -      n      )

Write the rule for the nth term given #2,-4,8,-10,14,...

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