Arithmetic Progressions Class 10 Notes Maths Chapter 5

Students can go through AP 10th Class Maths Notes Chapter 5 Arithmetic Progressions to understand and remember the concepts easily.

Arithmetic Progressions 10th Class Notes Maths Chapter 5

→ Sequence : A sequence is a list of numbers following a specific pattern. A sequence
can have a finite or infinite number of members.
Example : 1, 2, 3, 4, 5, ……. is an infinite sequence of natural numbers.

→ Series : A series is the sum of the elements in the corresponding sequence.
Example : 1 + 2 + 3 +4 + 5….is the series of natural numbers.

→ Term : Each number in a sequence or a series is called a term. In general, a finite
sequence is represented as a₁, a₂, a₃, ……… a_n,
where 1, 2, 3, ……, n represents the position of the term.
In general, a finite series is represented a₁ + a₂ + a₃ +…. + a_n.
An infinite sequence represented a₁, a₂, a₃, a₄, …… and the infinite series is represented as a + a + a

→ Progression : A progression is a sequence in which the general term can be can be expressed using a mathematical formula.

→ Arithmetic Progression: An arithmetic progression (AP) is a 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. Here the fixed number is called the common difference. As it is the difference between any two consecutive terms, for any A.P, if the common difference may be :

1. If d > 0, Positive, the AP is increasing.
2. If d = 0, Zero, the AP is constant.
3. If d < 0, Negative, the A. Pis decreasing.

In an arithmetic progression, the first term t₁ is represented by the letter “a”, the last term tn is represented by “l”, the common difference between two terms is represented by “d”, and the number of terms is represented by the letter “n”.
t₁ = a, t_n = 1, common difference = d
Therefore, the standard form of the arithmetic progression A.P. is a, a + d, a + 2d, a + 3d, a + 4d, …….. a + (n – 1)d

→ General term/The nth Term of an AP : The nth term of an A.P is given by T_n = a + (n – 1)d, where a is the first term, d is the common difference and n is the number of terms.
t_n = a + (n – 1)d
An arithmetic progression may have finite or infinite terms.
Finite A.P. : 12, 24, 36, 48, 60
Infinite A.P. : 1, 2, 3, 4, 5, 6, 7, ………

→ Sum of Terms in an AP: The sum to n terms of an A.P is given by: S_n = n/2{2a + (n – 1)d} Where a is the first term, d is the common difference and n is the number of terms. The sum of n terms of an A.P is also given by S_n = n/2(a + 1)

→ Sum of n Terms of AP Proof : Let’s consider the generalized representation of Arithmetic Progression, the sum of all the terms in an A.P. is a, a + d, a + 2d, a + 3d, a + 4d, …………. a + (n – 1)d
S_n = (a + a + d+ a + 2d + a + 3d + a + 4d + …………… a + (n – 1)d) → (i)
Rewriting the above equation in reverse order we get,
Sn = (a + (n – 1)d + a+ (n – 2)d + a + (n – 3)d+ ……… +a) → (ii)
Adding equation (i) and equation (ii)
2S_n = (2a + (n – 1)d + 2a + (n – )d + ……… + 2a + (n – 1)d) (n terms)
2S_n = [2a + (n – 1)d] x n
S_n = \(\frac{n}{2}\)[2a + (n – 1)d]

→ Mind map:

First term	a
Common difference	d
General form of AP	a, a + d, a + 2d, a + 3d,….
nth term	an = a + (n – 1)d
Sum of first n terms	Sn = (n/2) [2a + (n – 1)d]
Sum of all terms of AP	S = (n/2)(a + l) n = Number of terms l = Last term

→ Formulae :

Conditions		Formulae
Sum of n terms of AP when the first term is a		Sn = n/2[a+(n – 1)d]
Sum of n terms of AP when the first term is a and the last term is l		Sn = n/2[a + l]
Sum of first n Natural Number		Sn = n(n + 1)/2
Sum of square of first n Natural Number		Sn = n(n + 1)(2n + 1)/6
Sum of cube of first n Natural Number		Sn = [n(n + 1)/2]2