# 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 a1, a2, a3, a4,...... Then, the first term is a1, second term is a2, third term is a3 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 a1, a2, a3 ,..... is an AP, if the difference a2 – a1, a3, give the same value.

3. nth term of an AP : The nth term an (or the general term) of an AP is an = 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 an (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
$$s_n=\frac{n}{2}[a+l]$$