Answer: Arithmetic , Divergent
Step-by-step explanation:
Given sequence is
{aₙ} = { 4 , 10/3 , 8/ 3 , 2 , . . . }
To check whether the sequence is geometric or not , we divide second term by first term to find the common ratio . Then we again divide third term by second term to get common ratio .
The common ratio we get would same , then it is geometric .
10/3 10 5
r₁ = ----------- = ------ = ------
4 12 6
8/3 8 3 4
r₂= ---------- = ---------- * ------------ = -----
10/ 3 3 10 5
Thus the common ratio are not same . So the sequence is not geometric .
Now we check for arithmetic .
We take difference of second and first term and then difference of third and second term . If it will be same then it is arithmetic . This is called common difference , d .
10 - 2
d₁ = ------ - 4 = -------
3 3
8 10 - 2
d₂ = ------ - -------- = ---------
3 3 3
Thus the common difference is same .
So the given sequence is arithmetic .
To find whether it is convergent or divergent , we need to write sum of n terms first .
Formula for finding sum of n terms of arithmetic sequence is
n
sₙ = ----- [ 2a + ( n - 1 ) d]
2
We have a = 4 , d = - 2/3 .
Plug in this formula we get
n n 2 2
sₙ = ------- [ 2 * 4 + ( n - 1 ) ( -2/3) ] = ------ [ 8 - ----- n + ------ ]
2 2 3 3
n 26 2
sₙ = ------ [ ------ - ------- n ]
2 3 3
To check whether it is convergent or divergent , we take limit sₙ approaches to infinity .
n 26 2
lim sₙ = lim { --- [ --- - ------ n ] } = - ∞
n → ∞ n→∞ 2 3 3
As the sequence diverge , thus the series is divergent .
Thus given series is arithmetic , divergent .
Second is the correct option .