Classify the series as arithmetic or geometric then determine whether the series is convergent or divergent

Geometric , convergent

Step-by-step explanation:

Given sequence is    {  12 + ( -8 )  + 16/3 + ( - 32/9 ) + 64/27 + . . .  . . . . }

To check whether the sequence is arithmetic , we first find difference of first two terms then find difference of third and second term .

If we get both the difference same , then it is arithmetic .

d₁   =   - 8 - ( 12 ) =  - 20

           16                              40

d₂  =       - ( - 8 )  =    

             3                                3

Common difference is not same , thus it is not arithmetic .

To check whether sequence is geometric , we divide second by first term and then third by second term . If we get the same ratio , then it is geometric .

             -8              - 2

r₁   =   =  

             12                3

            16/3           16                     -2

r₂   =   =        =    

            - 8              3 * ( -8)              3

Thus common ratio is same , so it is geometric .

Now we need to check whether it is convergent or divergent .

We have an infinite geometric series .

It is convergent if | r | < 1 , that is common ratio is less than 1 .

We have | r |  = | - 2/3 |  = | - 0.66 |  = 0.66 < 1 .

Thus the  geometric series converges .

Thus given series is geometric , convergent .

Third is the correct option .