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  Theory of Arithmetic Sequences

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 - an

General 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).

1965_arithmetic sequence_1.jpg

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.

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