help please rsive rule and an explicit rule that will genera

help! please

rsive rule and an explicit rule that will generate the values in the table 17. Find both a recursive a recursive rule and an explicit rule that will generate the values in the table below, provided that al = 1 (2 Recursive rule: Explicit rule: 18. Find both a rec ursive rule and an explicit rule that will generate the values in the table below, provided that a 2 4 6 17 Recursive rule: Explicit rule:

Solution

18)

n              1             2           3           4            5          6

a_n            2             3            5          9            17         33

we can see that the difference between two consecutive (a_n) is

                      1           2              4            8           16

we can see that 1 = 2⁰ ,    2 = 2¹,     4 = 2²     ,8 = 2³    and 16 = 2⁴

and the a_n is \"1\" more then this difference

now to form a general formula for a_n

a₁ = 1 + 2⁰      =    1   +   2^(1-1)    =   1 + 2^(n-1)                 here n = 1 for a₁

a₂ = 1 + 2¹         =   1      + 2^(2-1)    =   1 + 2^(n-1)                       here n = 2

a₃ = 1 + 2²          =    1   +   2^(3-1)     =   1 + 2^(n-1)                        here n=3

and so on

therefore

a_n   = 1 + 2^(n-1)