Chapter: Arithmetic Progressions - Class 10

An arithmetic progression is a list of numbers where the difference between any two consecutive terms is always the same. This chapter introduces the concept of Arithmetic Progression (AP) as well as generic concepts like sequence and series and numerous AP formulas like the sum of n terms of AP, the nth term of an AP and so on.

Sequence

A sequence is an arranged set of numbers that follows a specific rule or pattern in a particular order.

For example: 2, 4, 6, 8, 10, 12, 14 and 16 is a sequence as each succeeding term is obtained by adding 2 to the preceding term.

Finite Sequence

A finite sequence is a list of numbers that has a specific and limited number of terms. It has a clear starting point and an ending point.

For example: 1, 3, 5, 7, 9 and 11 is a finite sequence.

Infinite Sequence

An infinite sequence is like a never-ending list of numbers. It goes on forever and doesn't have a specific endpoint.

For example 1, 3, 5, 7, 9, 11…… is an infinite sequence as it has no last term.

Series

A series is the sum of the terms in a sequence.

For example: Sequence 1, 4, 7, 10,…is expressed as a series is 1 + 4 + 7 + 10 +….

Arithmetic Progression (A.P.)

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is known as the "common difference". The common difference can be positive, negative or zero.

For example: 7, 11, 15, 19,............ is an arithmetic progression and the common difference is 4.

If three quantities a, b and c will be in A.P., then
b − a = c − b
2b = a + c

a_n or T_n denotes the n^th term of a sequence.

a, a + d, a + 2d, a + 3d, … is the general form of an A.P. where a is the first term and d is the common difference.

If there are a finite number of terms in an A.P., then it is called finite A.P.

If an A.P. does not have a last term, then it is called an infinite A.P.

n^th term of an A.P.

The nth term of an A.P. is denoted by a_n, T_n or l.
The formula to find the nth term of an A.P. is

a_n = a + (n − 1) d

Where,
an is the nth term.
a is the first term.
d is a common difference.
n is the number of terms.

Also,

Common difference d = T_n − T_{n − 1}

The nth term of an arithmetic progression (A.P.) is a formula used to find the value of any term in the sequence without listing all the terms.

Example: If the first term of an A.P. is 4 and the common difference is − 2, then what is the value of the sixth term?

a) − 4
b) − 8
c) − 2
d) − 6

Answer: d) − 6

Explanation: We know that a_n = a + (n − 1) d

Given: a = 4, d = − 2 and n = 6

a₆ = 4 + (6 − 1) (− 2)
= 4 + 5 (− 2)
= 4 − 10
= − 6

Hence, − 6 is the sixth term.

Sum of the first n terms of an A.P.

The sum of the first n terms of an arithmetic progression (AP) is often denoted by Sn. It represents the total value obtained by adding up the first n terms of the sequence.

The formula to find S_n is

S_n = n ÷ 2 [2a₁ + (n - 1)d]

Or

S_n = n ÷ 2 [a + l]

Where

S_n is the sum of the first n terms.

n represents the total number of terms.
a₁ (a) is the first term of the sequence.
d is a common difference.
l is the last term.

Also,

n^th term = Sum of n terms − Sum of (n − 1) terms

T_n = S_n − S_n − 1

Example: What is the sum of the first 20 terms of the A.P. 2, 5, 8, 11, …?

a) 610
b) 620
c) 630
d) 590

Answer: a) 610

Explanation: We know that S_n = n ÷ 2 [2a₁ + (n - 1)d]

Given:

n = 20
a = 2
d = 5 − 2 = 3

S₂₀ = 20 ÷ 2 [2(2) + (20 - 1)3]

= 10[4 + 19 3]
= 10[4 + 57]
= 10 × 61
= 610

Quick review of formulae related to A.P.

Arithmetic Mean

The arithmetic mean between two numbers is the number that you can put right in the middle so that it creates an arithmetic progression with those two numbers.

If a, b and c are in arithmetic progression, then
b− a = c − b
⇒    2b = a + c
⇒      b = (a + c) / 2

Where b is the arithmetic mean of a and c.

For example: If 6, 12 and 18 are in A.P., then 12 is the arithmetic mean of the A.P.

Some useful facts about an A.P.

If you adjust every term in an arithmetic progression (A.P.) by adding, subtracting and multiplying or dividing by the same number, the resulting sequence will also be an arithmetic progression.

To determine an odd number of terms in an arithmetic progression when the sum is known, choose 'a' as the middle term and 'd' as the common difference.
Thus,
Three terms may be taken as: a − d, a, a + d
Five terms may be taken as: a − 2d, a − d, a, a + d, a + 2d

To determine an even number of terms in an arithmetic progression, choose
a − d, a + d as the middle term and '2d' as the common difference.
Thus, four terms are taken as: a − 3d, a − d, a + d, a + 3d

If a, b, c are in A.P., then (a - b) ÷ (b - c) = 1