### Concept of Arithmetic Progression / Introduction to Arithmetic Progression / Basics of Arithmetic Progression

**Tags:**Definition^{th}standard**,**Arithmetic Progression Exercise, Practice Pages and Worksheet PDF for class 10, what is a sequence? Define common difference, General term, nth term for grade 10, How to find “d” using any two consecutive terms for class X, Introduction to Arithmetic SequencesArithmetic Progressions

**Introduction**We have observed many things in nature that follow a certain pattern, such as the holes of a honeycomb, the grains on a maize cob, the spirals on a pineapple etc.

We often observe interesting patterns of numbers in which quantity increases or decreases progressively in our day-to-day life. Some such examples are

**Example 1:**Ria applied for a job and got selected. She has been offered a job with a starting monthly salary of rupees 10,000 with an annual increment of rupees 1000 in her salary. Her salary (in rupees) for the 1st, 2nd, 3^{rd}…… years will be respectively.10,000, 11,000, 12,000, and so on

**Example 2:**In a saving scheme the amount becomes 2 times of itself after every 3 years. The maturity amount (in rupees) of an investment of rupees 10,000 after 3, 6, 9 and 12 years will be, respectively: 20,000, 40,000, 80,000, and so onIn the examples above, we observed that the succeeding terms are obtained by adding a fixed number in the first example and by multiplying with a fixed number in the second example. These patterns are generally known as

__sequences__.**Therefore,**a**sequence**is an arrangement of numbers in a definite order according to some rule.__Example:__2, 4, 6, 8, 10… Here numbers are even natural number.Now we will study a particular type of sequences which are known as

__arithmetic progressions__.**Arithmetic Progression:**An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.**Example 1:**-3, -2, -1, 0 …..Each of the numbers in the list is called a

**term**. We denote the terms of a sequence by**a**….etc. or_{1}, a_{2}, a_{3}**x**…etc. Here, the_{1}, x_{2}, x_{3}__subscripts denote the positions__of the terms. The__first term__of the A.P is denoted by**a**and the number at the second place is called the_{1}__second term__and is denoted by**a**and so on. In general, the number at the nth place is called the_{2}__nth term__of the sequence and is denoted by**a**. The term is called the_{n}__general term__of the sequence. In the above example a_{1 }(first term) = -3, a_{2 }(second term) = -2, a_{3}(third term) = -1, a_{4}(fourth term) = 0, and so on.Each term is obtained by adding a

__fixed number__except first term. Here in the above example each term is obtained by adding 1 to the term preceding it. And we can write the other next term by adding same fixed number. This fixed number is called the**common difference**of the A.P (Arithmetic Progression). It can be__positive, negative or zero__. Remember that we can find**d**using any two consecutive terms, once we know that the numbers are in AP. In the above example**d**= a_{n}–a_{n-1}(difference between the next term and previous term of the A.P.) i.e. a_{2}– a_{1 }=_{ }-2 – (-3) = 1**Example 2:**Let us consider the sequence of even natural numbers 2, 4, 6, 8, 10 …..Here we have,

**a**(first term) = 2 = 2x_{1}**1****a**(second term) = 4 = 2x_{2}**2****a**(third term) = 6 = 2x_{3}**3****a**(fourth term) = 8 = 2x_{4}**4****a**(fifth term) = 10 = 2x_{5}**5**and so on.It is evident from this that

**a**_{n}_{ }(nth term) = 2x**n**= 2**n**(where n = 1, 2, 3…..)^{}**Example 3:**Let us now consider the sequence of squares of natural numbers i.e., 1, 4, 9, 16, 25 ……Here we have,

**a**(first term) = 1 =_{1}**1**^{2}**a**(second term) = 4 =_{2}**2**^{2}**a**(third term) = 9 =_{3}**3**^{2}**a**(fourth term) = 16 =_{4}**4**^{2}**a**(fifth term) = 25 =_{5}**5**^{2}and so on.It is evident from this that

**a**_{n}_{ }(nth term) =**n**^{2 }(where n = 1, 2, 3…..)^{}**Example 4:**Let us study the sequence of odd natural numbers i.e., 1, 3, 5, 7, 9 ….Here we have,

**a**(first term) = 1 = 2x_{1}**1**-1**a**(second term) = 3 = 2x_{2}**2**-1**a**(third term) = 5 = 2x_{3}**3**-1**a**(fourth term) = 7 = 2x_{4}**4**-1**a**(fifth term) = 9 = 2x_{5}**5**-1 and so on.It is evident from this that

**a**_{n}_{ }(nth term) = 2x**n**-1 = 2n-1 (where n = 1, 2, 3…..)^{}We can also describe a sequence by writing the

__algebraic formula for its nth term or general term.__In some cases, the terms of the sequence do not follow some fixed pattern but they are generated by some__recursive relation__.**Example 5:**Consider for instance, the sequence, 1, 2, 3, 5, 8 …..Here we have,

**a**(first term) = 1_{1}**a**(second term) = 2_{2}**a**(third term) = 2 = 1+2 = a_{3}_{1}+ a_{2}**a**(fourth term) = 3 = 1+3 = a_{4}_{2}+ a_{3}**a**(fifth term) = 5 = 2+3 = a_{5}_{3}+a_{4}and so on.It is evident from this that

**a**_{n}_{ }(nth term) = a_{n-1 }+ a_{n-2 }for n>2**Example 6:**For the AP: 3, 1, -1, -3…. write the first term a_{1}and the common difference d.Here, first term (

**a**) = 3_{1}Common difference (

**d**) = a_{n}–a_{n-1}= a

_{2}– a_{1}= 1-3

= -2

**Example 7:**Is 4, 10, 16, 22 ….an A.P? If they form an A.P, write the next two terms.Here,

**d**= a_{2}– a_{1}= 10 – 4 = 6**d**= a_{3}– a_{2}= 16 – 10 = 6**d**= a_{4}– a_{3}= 22 – 16 = 6We can see that all the value of common difference is fixed or equal.

Hence, 4, 10, 16, 22….an A.P and next two terms are 22 + 6 = 28 and 28 + 6 = 32.

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