18 Find both a recursive rule and an explicit rule that will

18. Find both a recursive rule and an explicit rule that will generate the values in the table below, provided that ao 33 Explicit rule: Recursive rule:

Solution

we can see that

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 write 1 = 2⁰ ,    2 = 2¹,     4 = 2²     ,8 = 2³    and 16 = 2⁴    and so on

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

now to form a general formula for a_n

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

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

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

and so on

therefore

a_n   = 1 + 2^(n-1)