CBSE Class 10 Maths Revision Notes for Arithmetic Progression

—  A sequence is an arrangement of numbers in a definite order according to some rule.
—  The various numbers occurring in a sequence are called its terms.
—  We denote the terms of a sequence by a₁, a₂, a₃…etc. Here, the subscripts denote the positions of the terms in the sequence.
—  In general, the number at the n^th place is called the n^th term of the sequence and is denoted by an. The nth term is also called the general term of the sequence.
—  A sequence having a finite number of terms is called a finite sequence.
—  A sequence which do not have a last term and which extends indefinitely is known as an infinite sequence.

—  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.
—  Each of the numbers of the sequence is called a term of an Arithmetic Progression. The fixed number is called the common difference. This common difference could be a positive number, a negative number or even zero.

—  The general form of an A.P. is a, a + d, a + 2d, a + 3d….., where ‘a’ is the first term and ‘d’ is the common difference.
—  The general term(n^th term) of an A.P is given by an = a + (n - 1)d, where ‘a’ is the first term and ‘d’ is the common difference.

—  If the A.P a, a + d, a + 2d, …….., l is reversed to l, l – d, l – 2d, ….., a then the common difference changes to negative of the common difference of the original sequence.
—  To find the n^th term from the end, we consider this AP backward such that the last term becomes the first term.

1, (1 - d), (1 - 2d) ……
The general term of this AP is given by a_n = ℓ + (n - 1)(-d)

When we are given an algebraic formula for the general term of the sequence:

Step 1: Obtain an.
Step 2: Replace n by (n + 1) in an to get an+1
Step 3: Calculate an+1 – an
Step 4: Check the value of a_n+1 – an.
If a_n+1 – an is independent of n, then the given sequence is an A.P. Otherwise, it is not an A.P.

OR

A list of numbers a₁, a₂, a₃…… is an A.P, if the differences a₂ – a₁, a₃ – a₂, a₄ – a₃ … give the same value, i.e., a_k+1 – a_k is same for all different values of k.

If three number a, b, c (in order) are in A.P. Then,
b – a = c – b = common difference
⇒ 2b = a + c

Thus a, b and c are in A.P., if and only if 2b = a + c. In this case, b is called the Arithmetic mean of a and c.

—  Sum of n terms of an A.P. is given by 

     
     where ‘a’ is the first term, ‘d’ is the common difference and ‘n’ is the total number of terms.

—  Sum of n terms of an A.P. is also given by:

     
    where ‘a’ is the first term and ‘λ’ is the last term.

—  Sum of first n natural numbers is given by  .