Answer: 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 .
