Arithmetic sequences

The second generator is an arithmetic sequence method that generates random number using the following mathematical operation ... 

A sequence of numbers is a list of numbers created by a particular pattern or mathematical rule. An arithmetic sequence is a list of numbers in which there is a common difference between the consecutive numbers in the sequence. So consecutive integers are a special type of arithmetic sequence. The rule that allows you to add up any number of terms in an arithmetic sequence also lets you solve some problems involving the sums of consecutive integers. 

Before applying the rule for the sum of a certain number of terms in an arithmetic sequence, you need to be able to find the nth term in a sequence when... 

The nth term of an arithmetic sequence may be the fourth term or the tenth term or any number of term. You don t want to have to list the first 99 terms in order to find the hundredth term, so the following rule comes in handy. 

The nth term of an arithmetic sequence is an, which is equal to + d(n - 1), where ax is the first term and d is the difference between each of the terms in the sequence. 

The Problem If the first term of an arithmetic sequence is 3 and the 50th term is 297, then what is the common difference between the terms ... 

The generalized formula for the sum of any number of terms of an arithmetic sequence allows you to add up the terms no matter where you start and where you stop in the sequence. 

The sum of n terms of an arithmetic sequence is equal to half n times the sum of the first and last terms of the sequence. 

The Problem If the sum of the ten terms in an arithmetic sequence is 1,135 and if the difference between the terms is 3, then what are the first and last terms ... 

The formula for finding the sum of a list of consecutive integers requires that you have the first and last terms in the list. The multiples of 4 are all four units apart. You have an arithmetic sequence with terms the difference of which is 4. (You can find more on arithmetic sequences in Setting the stage for the sums, earlier in this chapter.) So, to find the 20th term in the list of multiples of 4 that start with 60, use the formula an = + d(n - 1), giving you... 

Beeler defined the broad scope of computer experiments as follows Any conceptual model whose definition can be represented as a unique branching sequence of arithmetical and logical decision steps can be analysed in a computer experiment... The utility of the computer... springs mainly from its computational speed. But that utility goes further as Beeler says, conventional analytical treatments of many-body aspects of materials problems run into awkward mathematical problems computer experiments bypass these problems. 

When the point values are average probabilities, the overall result from combining system.s as combinations of sequences and redundancies is found by simply combining the mean probabiliiies according to the arithmetic operations. 

The solution of the previous equations require careful attention to the sequence of the arithmetic. Perhaps one difficult requirement is the need to establish the L or G in Ib/hr/ft of tower cross-section, requiring an assumption of tower diameter. The equations are quite sensitive to the values of A and B. 

Recently, Teymour and coworkers developed an interesting computational technique called the digital encoding for copolymerization compositional modeling [20,21], Their method uses symbolic binary arithmetic to represent the architecture of a copolymer chain. Here, each binary number describes the exact monomer sequence on a specific polymer chain, and its decimal equivalent is a unique identifier for this chain. Teymour et al. claim that the... 

A closely related problem was considered in the seventeenth century. It is the famous arithmetic triangle developed by Pascal that is shown in Table 4. It is constructed by writing the number one twice as the first line. The first column is then filled with an infinite sequence of the number one. Subsequently, each value in the table is calculated by taking die sum of the number immediately above and the number to the left of the latter. It is then apparent that the second column is given by C(n, 1) = n, the third by C(n, 2) = n n — l)/2, and in general as given by Eq. (5). 

Equations 4-4, 4-5, and 4-6 represent the three elements of the matrix product of [A] and [B], Note that each row of this resulting matrix contains only one element, even though each of these elements is the result of a fairly extensive sequence of arithmetic operations. Equations 4-4, 4-5, and 4-7, however, represent the symbolism you would normally expect to see when looking at the set of simultaneous equations that these matrix expressions replace. Note further that this matrix product [A][fl] is the same as the entire left-hand side of the original set of simultaneous equations that we originally set out to solve. 

Many calculators can do positive and negative arithmetic. It is a good idea for you to be able to do this arithmetic mentally, but following is the key sequence for most calculators when doing positive and negative arithmetic. 

If your calculator has the positive/negative key, then it will perform positive and negative arithmetic. Most calculators have you enter the sign of a negative number after you enter the magnitude. So -8 would be entered as filFS. To perform the problem -3 —6, you would use the key sequence to get the result of 3. 

If the test you will be taking allows the use of a fractional calculator, you may want to invest in one. This can save precious time and increase accuracy when taking a timed test. If your calculator has a key that looks like 0, then it does fractional arithmetic. Most calculators will follow the key sequence as described below. Check your calculator with this example To perform the operation 4- - 7, enter the key sequence ... 

In the punched-card machines, if a calculation involved a sequence of many arithmetic operations, the machines were set up for one of these operations, which was performed on the data punched in as many cards as might be required. The machines were then... 

We can also consider the number of boundaries for objects in different dimensions. A line segment has two boundary points. A square is bounded by four line segments. A cube is bounded by six squares. Following this trend, we would expect a hypercube to be bounded by eight cubes. This sequence follows an arithmetic progression (2, 4, 6, 8. ..). 

One value in the sequence, 7, is missing because no possible integral values exists for (l/+] +f)=7. Other higher missing values exist where (1 +1 +f) cannot be an integer 15, 23, 28, etc., but note that this is only an arithmetical phenomenon and is nothing to do with the structure. 

Instructions can be modified by the computer. It is frequently found that the same instruction (or group of instructions) can be used repetitively if changed in some simple way. Since instructions are available in storage and can be subjected to the same arithmetic operations as any other numerical data, the computer can modify its instructions as the calculation proceeds. This permits relatively short sequences of instructions to be equivalent to much longer sequences on punched-card calculators. 

The Formulation of the Electronegativity Scale.—In Section 3-4 it was pointed out that the values of the difference between the energy D A—B) of the bond between two atoms A and B and the energy expected for a normal covalent bond, assumed to be the arithmetic mean or the geometric mean of the bond energies Z>(A—A) and Z>(B—B), increase as the two atoms A and B become more and more unlike with respect to the qualitative property that the chemist calls electronegativity, the power of an atom in a molecule to attract electrons to itself. Thus both A, the deviation from the arithmetic mean, and A, the deviation from the geometric mean, increase rapidly in the sequence HI,... 

---