**Arithmetic progression (A.P)**

**An Arithmetic progression is defined as that in which the difference between any two terms is always constant (fixed).**

** Or**

**Any quantity is increase or decrease by the same constant then such quantities form a series which is known as an Arithmetic Progression (A.P).**

**This constant is known as a common difference of the Arithmetic Progression (A.P).**

**So , we can say that Arithmetic Progression (A.P) is increase or decrease by a fixed number ( common difference) .**

**nth term of an Arithmetic Progression (A.P).**

** Let **

** a = 1st term**

** d = common difference**

**Formation of an Arithmetic Progression(A.P).**

**1st term = a **

**2nd term = a + d**

**3rd term = a + 2d**

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**nth term = a + (n-1)d**

**Tn = a + (n-1)d**

**nth term from last :**

**Let l be the last term of an arithmetic progression (A.P)**

**Tn = a + (n-1)d = l**

**Common difference :**

** d = Tn – Tn-1**

** = a + (n-1)d – [a + (n-2)d] **

**= -d**

**Then , nth term from last**

** T’n = l – (n-1)d**

**Sum of First n terms of an A.P**

** Let Sn be the summation of first n terms of an A.P**

**Sn = a +(a+d) + (a+2d) + **

** ( a+3d)+…+ (a + (n-1)d ).**

**This can be also written as**

**Sn = (a + (n-1)d )+(a+(n-2)d) **

** +… + (a+d) + a.**

**Add both equation—**

**2Sn =2a + (n-1)d + 2a + (n-1)d +**

**2a + (n-1)d + …… n terms**

**Sn = n/2[2a + (n-1)d]**

**Note :**

** Tn = Sn – Sn-1**

**Important points : **

**If a1 , a2 , a3 , …… are in A.P.**

**a) Addition (+): **

** Add a number k in each terms of an A.P. , then **

**a1 + k , a2 + k , a3 + k , … are also in A.P.**

**b) Subtraction (—) : **

** Subtract a number k in each terms of an A.P. , then **

**a1 – k , a2 – k , a3 – k , … are also in A.P.**

**c) Multiplication ( x) : **

** Multiply a number k in each terms of an A.P. , then **

**a1 x k , a2 x k , a3 x k , … are also in A.P.**

**d) Division ( / ): **

** Divide a number k in each terms of an A.P. , then **

**a1 / k , a2 /k , a3/k , … are also in A.P. ( k ≠ 0 ).**

**2. Construct a series with two A.P’s**

**If a1 , a2 , a3 , …… and b1 , b2 , b3 …. are two A.P’s.**

**Addition :**

** (a1 + b1 ) + (a2 + b2 ) + (a3 + b3 ) + ….. are also in A.P.**

**b) Subtraction : **

** (a1 – b1 ) + (a2 – b2 ) + (a3 – b3 ) + ….. are also in A.P.**

**c) Multiplication :**

** (a1 x b1 ) + (a2 x b2 ) + (a3 x b3 ) + ….. are not in A.P.**

**d) Division :**

** (a1 / b1 ) + (a2 / b2 ) + (a3 / b3 ) + ….. are not in A.P.**

**3 . If a1 , a2 , a3 , …… are in A.P.**

**a ) a1 + an = a2 + an-1 = a3 + an-2 = ……..**

**b ) ar = 1/2( ar-k + ar+k ) , 0 ≤ k ≤ n-r**

**4. Three numbers are taken in A.P .**

** a – d , a , a + d. **

**similarly , five numbers are taken in A.P .**

** a – 2d , a-d , a , a+d , a + 2d . etc .**

**In general —**

**(2r + 1) numbers are taken in A.P.**

**a – rd , a-(r-1)d … , a -d , a , a + d , …, a + (r-1)d , a+rd.**

**5. Four numbers are taken in A.P .**

** a – 3d , a – d , a+d , a+3d ;**

**similarly , six numbers are taken in A.P .**

**a – 5d , a-3d , a-d , a+d , a+3d , a+5d; **

**In general —**

** 2r numbers are taken in A.P.**

**a – (2r-1)d , a-(2r-3)d ,….. , a-3d , a-d , a+d , a , a+3d , …, a +(2r-3)d , a + (2r-1)d .**

**6. Arithmetic Progression is a linear expression in n.**

**Ex. an + b.**

**7. Sum of n terms of any sequence is quadratic expression in n , then sequence in A.P.**

**Ex . a n^{2} + bn + c**

**7. Common difference of an A.P may be zero , positive or negative.**

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