# NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions

**Important Points**

**1. Sequence :**An arrangement of numbers in a definite order according to some rule is called a sequence. e.g., 1, 2, 3, 4, 5, 6, ... 2, 4, 6, 8, 10, ... 2, 4, 8, 16, 32, ... Sequence is said to be finite or infinite accordingly it has a finite or infinite number of terms. Consider a sequence a

_{1}, a

_{2}, a

_{3}, a

_{4},...... Then, the first term is a1, second term is a

_{2}, third term is a

_{3}and so on. Its nth term can be written as an which is known as the general term of the sequence.

**2. Arithmetic Progression :** A succession of number is said to be an Arithmetic Progression (AP), if each term of the sequence can be obtained by adding a fixed numbered to the preceding term except the first term a. The fixed numbered may be positive, negative or zero and is called the common difference of AP.

The

general term of an AP is a, a + d, a + 2d, a + 3d,...**Examples :**

(i) 3, 7, 11, 15, 19, ...

(ii) 15, 12, 9, 6, 3, ...

(iii) 2, 2, 2, 2, ....

(iv) – 2.5, – 3, – 3.5, – 4, – 4.5, ....

Note : A given list of numbers a_{1}, a_{2}, a_{3} ,..... is an AP, if the difference a_{2} – a_{1}, a_{3}, give the same value.

**3. nth term of an AP :** The nth term an (or the general term) of an AP is a_{n} = a + (n – 1)d, where a is the first term and d is the common difference.

**4. nth term of an AP from the End :** The nth term a_{n} (or the general term) of an AP from the end an = l – (n – 1)d, where l is the last term and d is the common difference.

**5. Sum of n terms of an AP :** The sum Sn of the first n terms of an AP is given by

$$s_n=\frac{n}{2}[2a+(n-1)d]$$**Note :** If l is the last term of the finite AP say the nth term, then the sum of all terms of the AP is given by

$$s_n=\frac{n}{2}[a+l]$$