1. John wants to fence a 150 square meters rectangular field. He wants the length and width to be natural numbers {1,2,3,...}. What field dimensions will require the least amount of fencing?

2. List the terms that complete a possible pattern in each of the following, and classify each sequence as arithmetic, geometric, or neither:
(a) 38, 33, 28, 23, 18, ____, ____, ____
(b) 640, 320, 160, 80, ____, ____, ____

3. Find the following sum: 6 + 8 + 10 + 12 + ... + 100.

4. A special rubber ball is dropped from a height of 128 meters. Each time the ball bounces to half of its previous height. After 1 bounce the ball reaches a height of 64 meters.
(a) How high did the ball reach after the 7th bounce?
(b) How high does the ball reach after the nth bounce?

5. Use 3 and 4 as the first two terms of a Fibonacci-type sequence.
Find the next five terms:
3, 4, _______, _______, _______, _______, _______

6. Solve the following equations for x.
(a) 3x + 5 = 23
(b) 6(x - 8) = 2x + 12

(a) Prove this operation:
Let {xn} and {yn} be convergent sequences.
The sequence{zn} where zn:=xn-yn converges and lim (xn-yn)=lim zn=lim xn-limyn
What I attempted was this:
Suppose {xn} and {yn} are convergent sequences and write zn:=xn-yn. Let x:=lim xn, y:=lim yn and z:=x-y
Let epsilon>0 be given. Find M1 s.t. for

Use electronic resources available to predict native disorder and inherent structure in proteins and RNA. Use IUPred for native protein disorder and the RNAfold web server to predict RNA secondary structure.
1.) Generate a random DNA sequences of 60 nucleotides using the site: http://www.faculty.ucr.edu/~mmaduro/random.ht

Please show all the work.
(a) Use the definition of a null sequence to prove that the sequence {an} given by
an = (-1)^(n+1) / (n^3 - 3) , n = 1, 2, ...
is null.
(b) Determine whether or not each of the attached sequences {an} converges. Find the limit of each convergent sequence.

Establish the convergence and find the limits of the following sequences
((1+1/2n)^n))
((1+1/n^2)^(2(n^2)))
((1+2/n)^n))
Give an example of a convergent sequence Xsubn of positive numbers with lim(Xsubn^1/n)=1
Give an example of a divergent sequence Xsubn os positive numbers with lim(Xsubn^1/n)=1

For Xsubn given by formulas establish either the convergence or the divergence of the sequence X=Xsubn
Xsubn= n/(n+1)
Xsubn= 2n^2+3/(n^2+1)
find the limits of the following sequences:
lim((2+ 1/n)^2))
lim((-1)^n/(n+2))
lim((n^1/2)-1)/((n^1/2)+1))
lim((n+1)/(n(n^1/2)))
establish the convergence and find the limits o

Suppose the sequences {a_n}_n and {b_n}_n are both bounded above.
a) Prove that for all n in the naturals sup{a_k + b_k: k>/=n} is less than or equals sup{a_k:k>/=n} + sup{b_k:k>/=n}
b) Use this to conclude: limsup (a_n + b_n) is less than or equals limsup(a_n) + limsup(b_n)
(all limits are n--> infinity)

If {sn}∞ n=1 is a sequence of real numbers such that sn ≤ M for all n and lim n--> ∞ sn =L; prove that L ≤ M. Is the statement true if we replace both inequalities with "<"?
See attachment for full equation.