Introduction
While arranging numbers in order, sometimes, we might encounter that the difference between any two consecutive numbers is always equal throughout the arrangement. In such cases, the ordered set of numbers is considered to be in an Arithmetic Progression, abbreviated as AP. It is a broad term that includes Arithmetic Sequence and Arithmetic Series within itself.
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Progression: Sequence and Series | Introduction
Example:
$$1,3,5,7,9$$
$$2,6,10,14,18$$
In each of the above-mentioned arithmetic progressions, the terms or numbers differ from their preceding by a constant difference, called the common difference of the progression and it is denoted by 'd'.
Mathematically, a progression is said to be an arithmetic progression if, for each term in the progression, there exists a common difference, d which is defined as $$\text{d = t}_\text{n} \text{ - t}_\text{n-1}$$.
General Term of an Arithmetic Progression
From the concept of common difference, we can consider the following arrangement to be in arithmetic progression provided that a and d are numbers.
$$\text{a},\text{a+d},\text{a+2d},\text{a+3d},...$$
Let us assume that the above progression has n terms then, it should be noted that the nth term in the progression has an additional term equal to the product of (n-1) and d, as compared to the first term 'a'.
Hence, it can be concluded that the nth term of the progression can be denoted by $$\text{a + (n-1)d}$$ which is also the formula for the last term of the progression.
Properties of Arithmetic Progression
Let us consider an arithmetic progression as follows:
$$\text{a}_0,\text{a}_1,\text{a}_2,\text{a}_3,...$$
Let k $\in$ R and k $\neq$ 0 then numbers in the above progression obey the following properties:
- If k is added to or subtracted from each term in the progression then the new progression is again in Arithmetic Progression.
$$\text{a}_0\text{ + k},\text{a}_1\text{ + k},\text{a}_2\text{ + k},\text{a}_3\text{ + k},...$$
$$\text{a}_0\text{ - k},\text{a}_1\text{ - k},\text{a}_2\text{ - k},\text{a}_3\text{ - k},...$$ - If each term is multiplied or divided k then the new progression is again in Arithmetic Progression.
$$\text{ka}_0,\text{ka}_1,\text{ka}_2,\text{ka}_3,...$$
$$\dfrac{\text{a}_0}{\text{k}},\dfrac{\text{a}_1}{\text{k}},\dfrac{\text{a}_2}{\text{k}},\dfrac{\text{a}_3}{\text{k}},...$$