Class 10 Mathematics · Chapter 5 NotesArithmetic Progressions

In nature and daily life, we often see patterns: petals of a sunflower, holes of a honeycomb, or the decreasing lengths of a ladder's rungs. An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first. This fixed number is called the common difference, denoted by d. It can be positive, negative or zero. For example, the list 8000, 8500, 9000, ... is an AP with first term 8000 and common difference 500. The general form of an AP is a, a + d, a + 2d, a + 3d, ... where a is the first term. An AP can be finite (having a last term) or infinite (without a last term). To define an AP, you need both the first term a and the common difference d.

• APeach term is obtained by adding a fixed number d to the preceding term
• Common difference d = aₖ₊₁ – aₖ, can be positive, negative or zero
• General forma, a + d, a + 2d, a + 3d, ...
• Finite AP has a last term; infinite AP does not

To check whether a given list of numbers is an AP, find the difference between consecutive terms. If the difference is the same every time, the list is an AP. For example, in 6, 9, 12, 15, ..., the differences are 3, 3, 3, so it is an AP with d = 3. In 1, 1, 2, 3, 5, ..., the differences are not the same, so it is not an AP. Remember to subtract the earlier term from the later term, even if the later term is smaller. For instance, in 6, 3, 0, –3, ..., we subtract 6 from 3 to get d = –3. Once you know a and d, you can write the AP by adding d repeatedly.

• Check if aₖ₊₁ – aₖ is the same for all consecutive terms
• d = a₂ – a₁ = a₃ – a₂ = ... = aₙ – aₙ₋₁
• If differences are not equal, the list is not an AP

The nth term (also called the general term) of an AP with first term a and common difference d is given by aₙ = a + (n – 1)d. This formula helps you find any term without writing all the previous terms. For example, if a = 2 and d = 5, then the 10th term is a₁₀ = 2 + (10 – 1) × 5 = 47. You can also use this formula to check whether a given number is a term of an AP. If solving for n gives a positive integer, the number is a term; otherwise, it is not. The formula is useful in many real-life situations, such as finding salary in a particular year or the number of terms in a sequence.

• aₙ = a + (n – 1)d
• aₙ is also called the general term
• If the last term is l, then l = a + (n – 1)d
• To check if a number is a term, solve for n; n must be a positive integer

The sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n – 1)d]. This formula is derived by adding the AP forward and backward, a method used by Gauss to sum numbers from 1 to 100. If the last term l is known, you can use the simpler form Sₙ = n/2 (a + l). These formulas involve four quantities: Sₙ, a, d and n. If you know any three, you can find the fourth. The sum formula is very useful for finding total savings, total production, total penalty, and many other cumulative totals.

• Sₙ = n/2 [2a + (n – 1)d]
• Sₙ = n/2 (a + l), where l is the last term
• Sum of first n positive integersSₙ = n(n + 1)/2
• aₙ = Sₙ – Sₙ₋₁

Arithmetic progressions appear in many practical situations. For example, a salary increasing by a fixed amount each year, the decreasing lengths of ladder rungs, the number of logs stacked in rows, or the total distance run in a potato race. By modelling these situations as APs, you can use the nth term and sum formulas to answer questions like: How many terms are needed to reach a certain sum? What is the total production over several years? How many rows are there if the top row has a certain number of items? These problems often require setting up equations using the AP formulas and solving for the unknown quantity.

• Model real-life situations as APs to find unknown terms or sums
• Use aₙ = a + (n – 1)d to find a specific term
• Use Sₙ = n/2 [2a + (n – 1)d] to find total sum
• Sometimes two values of n are possible, as in Example 13

If three numbers a, b, c are in AP, then b is called the arithmetic mean of a and c, and it is given by b = (a + c)/2. This means that the middle term is the average of the other two. For example, if 2, b, 8 are in AP, then b = (2 + 8)/2 = 5. The arithmetic mean is useful in problems where you need to find a missing term between two given terms of an AP.

• If a, b, c are in AP, then b = (a + c)/2
• b is the arithmetic mean of a and c