Question

Let a_{1}= - 1 , a_{n+1}= (6+a_{n}) /
(2+a_{n}).

a) Assume that the given recursive sequence is convergent. Find the limit.

b) Is the given sequence bounded? Why?

Answer #1

Consider the following.
a1 = 8, an + 1 = 9 − an
Calculate, to four decimal places, the first eight terms of the
recursive sequence.
Does it appear to be convergent?
Yes No
If so, guess the value of the limit. (If the quantity diverges,
enter DIVERGES.)
Assume the limit exists and determine its exact value. (If the
quantity diverges, enter DIVERGES.)
lim n→∞ an =
Consider the following.
a1 = 1, an + 1 =
square root (4an)...

A sequence has a recursive formula of an = (-2)n (an-1) for n≥2.
The fourth term a4 is 1,536.
a. Find the first term a1. (5 points)
b. Find the sixth term a6. (5 points)

Find a general term (as a function of the variable n) for the
sequence{?1,?2,?3,?4,…}={45,1625,64125,256625,…}{a1,a2,a3,a4,…}={45,1625,64125,256625,…}.
Find a general term (as a function of the variable n) for the
sequence {?1,?2,?3,?4,…}={4/5,16/25,64/125,256/625,…}
an=
Determine whether the sequence is divergent or convergent. If
it is convergent, evaluate its limit.
(If it diverges to infinity, state your answer as inf . If it
diverges to negative infinity, state your answer as -inf . If it
diverges without being infinity or negative infinity, state your
answer...

Using the recursion formula an+1 = (3an - 2)^1/2, with a1 = 4/3,
do the following by induction.
a. Show the sequence { an } is monotone increasing.
b. Show the sequence is bounded above by 2.
c. Evaluate the the limit of the sequence.

Let (a_n)∞n=1 be a sequence defined recursively by a1 = 1, a_n+1
= sqrt(3a_n) for n ≥ 1. we know that the sequence converges. Find
its limit.
Hint: You may make use of the property that lim n→∞ b_n = lim
n→∞ b_n if a sequence (b_n)∞n=1 converges to a positive real
number.

A sequence is defined by a1=2 and an=3an-1+1. Find the sum
a1+a2+⋯+an

Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤ (a1)2 + (a2)2 +
... + (an)2.
Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).
Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s...

1. Find the limit of the sequence whose terms are given by an=
(n^2)(1-cos(5.6/n))
2. for the sequence an= 2(an-1 - 2) and a1=3
the first term is?
the second term is?
the third term is?
the forth term is?
the fifth term is?

Give a recursive description of the sequence of increasing even
numbers 2, 4, 6, ...

2. Given the recurrence relation an = an−1 + n for n ≥ 2 where
a1 = 1, find a explicit formula for an and determine whether the
sequence converges or diverges

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