what is arithmetic sequence?

What is an arithmetic sequence?
Definition:
An arithmetic sequence is a sequence having a common difference between successive terms.
e.g. 3, 5, 7, 9, 11, 13, 15.
The common difference between successive terms is 2.

Formula:
Let the first term T(1) = a and the common difference = d.
Then, T(1) = a,
T(2) = a + d,
T(3) = a + 2d,

T(n) = a + (n-1)d.

The nth term in an arithmetic sequence is T(n) = a + (n-1)d.

In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13... is an arithmetic progression with common difference 2.

就咁睇呢段你可能唔太明
你可以去http://en.wikipedia.org/wiki/Arithmetic_sequence
有詳細的數據

In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13... is an arithmetic progression with common difference 2.
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence is given by:


圖片參考：http://upload.wikimedia.org/math/0/2/9/029db8d36d83883674e157f930cf479d.png

and in general


圖片參考：http://upload.wikimedia.org/math/7/5/e/75e8cd43dc5817d40487480b96a07447.png

Sum (arithmetic series)
The sum of the components of an arithmetic progression is called an arithmetic series
Calculating the value of an arithmetic series
The value of an arithmetic series consisting of n terms
圖片參考：http://upload.wikimedia.org/math/7/a/a/7aaf8239aa023dd5e116e045ea26ebc6.png
is given by


圖片參考：http://upload.wikimedia.org/math/9/a/5/9a5fb55b3d24acaa271a9e4eea85cd83.png

Intuitively, this formula can be derived by realizing that the sum of the first and last terms in the series is the same as the sum of the second and second to last terms, and so forth, and that there are roughly n / 2 such sums in the series. An often-told story is that Carl Friedrich Gauss discovered this formula when his third grade teacher asked the class to find the sum of the first 100 numbers, and he instantly computed the answer (5050).
A different way to get the result, that avoids the fuzziness of the previous method when the number of terms is odd, is to think in terms of averages. The value of the arithmetic series is the number of terms in the series times the average value of the terms. The average must be (a1 + an) / 2, since the values appear evenly spaced out around around this point on the real number line. Put another way,
圖片參考：http://upload.wikimedia.org/math/1/b/6/1b6f167cc456fd1e3589000ce811fd39.png
is constant and equal to (a1 + an) / 2, which corresponds to the fact that successively taking terms from opposite sides of the series gives a constant average, which therefore must be the average of all terms in the series.
Proof of the formula
Express the arithmetic series in two different ways:

圖片參考：http://upload.wikimedia.org/math/4/0/c/40c3b19d369440ef820a8e297933b48d.png


圖片參考：http://upload.wikimedia.org/math/a/c/7/ac7da0bc6ae0573006217e5f0c827709.png

Add both sides of the two equations. All terms involving d cancel, and so we're left with:

圖片參考：http://upload.wikimedia.org/math/1/2/2/122f1a6e41cb9737b9ffee4e154bae43.png

Rearranging and remembering that an = a1 + (n − 1)d, we get:

圖片參考：http://upload.wikimedia.org/math/1/3/4/134fa4927dc3bef1cd63f89449590a13.png
.
Arithmetic series and sigma notation
Arithmetic series are commonly expressed using sigma notation. As an example, the arithmetic series

圖片參考：http://upload.wikimedia.org/math/e/e/d/eed17e840b72f428e5f2708c6cb6f59f.png

can be more succinctly written using sigma notation as

圖片參考：http://upload.wikimedia.org/math/2/8/8/288b3dfab71e9b2c447bd1be6edb6f87.png

Likewise, an arithmetic series

圖片參考：http://upload.wikimedia.org/math/6/b/3/6b39dab7b3b0a0b04ffc083f44d2ebf8.png

can be written as

圖片參考：http://upload.wikimedia.org/math/d/5/0/d5014f8c09533c856f81ad07c99c9cb0.png