Arithmetic Sequences:
The arithmetic sequence is a sequence in which the difference among consecutive terms is constant.
Common Difference:
As this difference is common to all the consecutive pairs of terms, it is termed as the common difference. It is represented by d. When the difference in consecutive terms is not constant, then sequence is not arithmetic. The common difference can be determined by subtracting two consecutive terms of sequence.
The formula for common difference of an arithmetic sequence is: d = an+1 - anGeneral Term:
The arithmetic sequence is a linear function. Rather than y = mx+b, we write an = dn+c where d is the common difference and c is the constant (that is, not the first term of sequence, though).
The recursive definition, as each term is determined by adding the common difference to the prior term is ak+1 = ak + d.
For any term in sequence, we have added the difference one less time than the number of term. For illustration, for first term, we have not added the difference at all (that is, 0 times). For second term, we have added the difference once. For third term, we have added the difference two times.
The formula for general term of an arithmetic sequence is: an = a1 + (n-1) d
Partial Sum of an Arithmetic Sequence:
The series is a sum of a sequence. We wish to determine the nth partial sum or the sum of first n terms of sequence. We will represent the nth partial sum as Sn.
Let consider the arithmetic series S5 = 2 + 5 + 8 + 11 + 14. There is a simple way to compute the sum of an arithmetic series.
S5 = 2 + 5 + 8 + 11 + 14
The key is to switch the order of terms. Addition is commutative; therefore changing the order does not modify the sum.
S5 = 14 + 11 + 8 + 5 + 2
Now, add such two equations altogether.
2*S5 = (2+14) + (5+11) + (8+8) + (11+5) + (14+2)
Note that each of such sums on the right hand side is 16. Rather than writing 16 (that is, the sum of first and last terms) five times, we can write it as 5 * 16 or 5 * (2 + 14)
2*S5 = 5*(2 + 14)
Lastly, divide the entire thing by 2 to get the sum and not twice the sum.
S5 = 5/2 * (2 + 14)
We have purposely not simplified the 2+14 and hence you can see where the numbers come from. The sum would be 5/2 *(16) = 5(8) = 40.
Now, when we try to figure out where the various parts of that formula come from, we can conjecture regarding a formula for the nth partial sum. The 5 is as there were five terms, n. The 16 is the sum of first and last terms, a1 + an. The 2 is as we added the sum twice and will remain a 2. Thus, the sum of first n terms of an arithmetic sequence is Sn=n/2*(a1+an)
There is another formula which is sometimes employed for nth partial sum of an arithmetic sequence. This is obtained by replacing the formula for general term to the above formula and simplifying. The preferred technique is to go ahead and determine the nth term, and then just plug that number to the formula.
Sn = n/2 * (2a1 + (n-1) d)
Illustration:
Determine the sum from k = 3 to 17 of (3k-2).
The first term is determined by replacing k = 3 to 3k-2 to obtain 7. The last term is 3(17) - 2 = 49. There are 17 - 3 + 1 = 15 terms. Therefore, the sum is 15/2 * (7 + 49) = 15/2 * 56 = 420.
Note that there are 15 terms there. Whenever the lower limit of the summation is 1, there is a small problem figuring out what the number of terms is. Though, whenever the lower limit is any other number, it seems to provide people difficulty. No one would argue that when we went from 1 to 10, there are 10 numbers. Though, the difference between 10 and 1 is simply 9. Therefore, if you are finding out the number of terms, this is the upper limit minus the lower limit plus one.
Latest technology based Algebra Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Algebra help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Algebra, project ideas and tutorials. We provide email based Algebra help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Algebra. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Algebra Homework help and assignment help services. They use their experience, as they have solved thousands of the Algebra assignments, which may help you to solve your complex issues of Algebra. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
Genetic code and Gene expression tutorial all along with the key concepts of Control of Gene Expression, Gene Expression in Bacteria, Hormonal Control of Gene Expression, Introduction to Genetic Code, Nature of the Genetic Code and Features of Genetic Code
tutorsglobe.com fixation assignment help-homework help by online dark reactions tutors
www.tutorsglobe.com offers Pair of Straight Lines homework help, Pair of Straight Lines assignment help, online tutoring assistance, geometry mathematics solutions by online qualified tutor's help.
Aves tutorial all along with the key concepts of Features of Class Aves, Features of Subclass Archaeonithes, Class Subclass Neornithes, Superorder Odontognathae, Palaeognathae, Struthioniformes and Tinamiformes
tutorsglobe.com net present value assignment help-homework help by online capital budgeting and project planning tutors
www.tutorsglobe.com offers Data Modelling homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
The Inorganic Fertilizers tutorial all along with the key concepts of What is a fertilizer, Organic Fertilizer versus Inorganic Fertilizer, Examples of Inorganic Fertilizers, Types of Inorganic Fertilizers, Advantages and disadvantages of Inorganic fertilizers
tutorsglobe.com basic features of heredity assignment help-homework help by online concept of heredity and variation tutors
tutorsglobe.com bronchitis assignment help-homework help by online respiration tutors
in the second generation of computers that the concept of programming language, Central Processing Unit (CPU), memory, and input and output units were developed.
To combine the transmitted RF waves and the receiver an instrument is essential. The received signal is processed to several actions and at last audio is obtained.
tutorsglobe.com total cost assignment help-homework help by online cost and revenue tutors
Theory and lecture notes of Steady-State Response of Electrodes all along with the key concepts of Electrode Electrical Model, Frequency Response and Amplifier Input Impedance Requirements. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Steady-State Response of Electrodes.
Theory and lecture notes of Accuracy, Condition Numbers and Pivoting all along with the key concepts of accuracy, condition numbers and pivoting, linear algebra, effect of rounding, Row Pivoting. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Accuracy, Condition Numbers and Pivoting.
Avail top-rated Budget and Budgetary Controls Assignment Help and get assured top-notch grades by submitting a masterpiece.
1953110
Questions Asked
3689
Tutors
1443362
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!