Transcribed Image Text: Which of the followings are true? I. sinn 2n is a divergent sequence. 2=1 II. The sequence {V3 – 1, 8 – 2, V14 – 3, v24 – 4, /35 – 5,…} converges to 1. III. {r*} is convergent when r = -0,5. IV. Let a1 1 and an+1 /1+2a, for n=2,3,… Then an is a divergent sequence. O a. I, II O b. II, II O c. I, IV O d. II, IV O e. I, II
To determine which of the statements are true, let’s evaluate each option one by one:
I. The sequence sinn/n is a divergent sequence.
To determine if the sequence sinn/n is divergent, we can use the limit test.
Let’s consider the limit as n approaches infinity:
lim(n->inf) sinn/n
Using the limit definition of sine, we have:
lim(n->inf) sin(n)/n
We know that lim(n->inf) sin(n)/n = 0 (you can prove this using the squeeze theorem or L’Hopital’s rule). Therefore, the sequence sinn/n converges to 0, not diverging. So, statement I is false.
II. The sequence {V3 – 1, 8 – 2, V14 – 3, v24 – 4, /35 – 5,…} converges to 1.
To determine if the given sequence converges to 1, we can examine the pattern of the terms.
Looking at the sequence, we notice that the nth term is given by n^2 – n + 2.
Let’s take the limit as n approaches infinity to see if the sequence converges:
lim(n->inf) (n^2 – n + 2)
As n approaches infinity, the dominant term will be n^2. Therefore, the limit is positive infinity, not equal to 1. So, statement II is false.
III. {r*} is convergent when r = -0,5.
To determine if the sequence {r*} is convergent when r = -0.5, we need to define what {r*} represents.
If {r*} refers to the sequence {rn} where n is a positive integer, then we can evaluate the given statement.
For r = -0.5, the sequence would be {-0.5, -0.25, -0.125, -0.0625, …}.
As the absolute value of r is less than 1, the sequence {r*} converges to 0. So, statement III is true.
IV. Let a1 = 1 and an+1 = 1 + 2an, for n = 2, 3, … Then an is a divergent sequence.
To determine if the sequence defined by a1 = 1 and an+1 = 1 + 2an is divergent, we need to observe the pattern of the terms.
Using the given recurrence relation, we can find the first few terms of the sequence:
a1 = 1
a2 = 1 + 2(1) = 3
a3 = 1 + 2(3) = 7
a4 = 1 + 2(7) = 15
…
From the given terms, it appears that the sequence is growing exponentially. To verify this, let’s examine the ratio of consecutive terms:
a2/a1 = 3/1 =3
a3/a2 = 7/3
a4/a3 = 15/7
…
From this pattern, we can see that the ratio of consecutive terms is always greater than 1. As a result, the sequence is growing without bound and is divergent. So, statement IV is true.
In conclusion,
Statement III is true.
Statements I, II, and IV are false.
Therefore, the correct answer is: Option b. II, II.