CBSE Class 10 Maths Chapter 5: Arithmetic Progressions Notes

Key Formulas and Properties of Arithmetic Progressions

• An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant.
• The general form of an AP is a, a + d, a + 2d, a + 3d, ... where a is the first term and d is the common difference.
• The nth term (general term) of an AP is given by the formula: a_n = a + (n - 1)d.
• The common difference (d) can be found by subtracting any term from its succeeding term: d = a_{n+1} - a_n.
• The sum of the first n terms of an AP is given by two formulas: 1. S_n = n/2 [2a + (n - 1)d] 2. S_n = n/2 [a + l] (where l is the last term, i.e., a_n)
• If a, b, c are in AP, then 2b = a + c (b is the arithmetic mean of a and c).
• The n-th term from the end of an AP with m terms is a_{m-n+1} or l - (n-1)d.
• If terms are selected at regular intervals from an AP, the new sequence also forms an AP.

Essential Terms for Arithmetic Progressions

Arithmetic Progression (AP)

A sequence of numbers in which each term after the first is obtained by adding a fixed number to the preceding term.

First Term (a)

The initial number in an arithmetic progression, denoted by a_1 or simply a.

Common Difference (d)

The constant difference between any term and its preceding term in an AP. It can be positive, negative, or zero.

General Term (n-th Term)

The formula that allows you to find any term a_n in the sequence based on its position n, the first term a, and the common difference d. It is a_n = a + (n - 1)d.

Finite AP

An arithmetic progression that has a limited number of terms, with a definite last term.

Infinite AP

An arithmetic progression that continues indefinitely, having an unlimited number of terms.

Sum of n Terms (S_n)

The total sum of the first 'n' terms of an arithmetic progression.

Understanding the Concept of an Arithmetic Progression

An Arithmetic Progression (AP) is fundamentally a list of numbers where the difference between any term and its succeeding term remains constant throughout the sequence. This constant difference is known as the common difference, denoted by d. For instance, in the sequence 2, 5, 8, 11, ..., the first term a is 2. The difference between 5 and 2 is 3, between 8 and 5 is 3, and so on. Here, the common difference d is 3. Similarly, in 10, 8, 6, 4, ..., a = 10 and d = 8 - 10 = -2. The common difference can thus be positive, negative, or even zero (e.g., 5, 5, 5, 5, ... where d=0).

The general form of an AP is expressed as a, a + d, a + 2d, a + 3d, .... From this pattern, we can easily deduce the formula for the nth term, often called the general term, a_n. The first term is a + (1-1)d = a. The second term is a + (2-1)d = a + d. The third term is a + (3-1)d = a + 2d. Following this logic, the n-th term will be a + (n-1)d. This formula, a_n = a + (n - 1)d, is incredibly powerful as it allows you to find any term in an AP without having to list out all the preceding terms. Understanding this basic structure is key to solving a majority of problems in this chapter, including finding missing terms, determining if a number is part of an AP, or even setting up equations for more complex scenarios involving multiple APs. Being able to identify the first term (a) and calculate the common difference (d) are the initial and most critical steps in solving any AP problem. Pay close attention to the signs of d as it significantly impacts the nature of the progression.

Worked Examples for Arithmetic Progressions

• Example 1: Finding the nth term Q: Find the 10th term of the AP: 2, 7, 12, ... A: Here, a = 2, d = 7 - 2 = 5. We need a_10. Using a_n = a + (n - 1)d, a_10 = 2 + (10 - 1)5 = 2 + 9 \times 5 = 2 + 45 = 47. The 10th term is 47.
• Example 2: Finding the sum of n terms Q: Find the sum of the first 20 terms of the AP: 1, 3, 5, 7, ... A: Here, a = 1, d = 3 - 1 = 2, n = 20. Using S_n = n/2 [2a + (n - 1)d], S_20 = 20/2 [2(1) + (20 - 1)2] = 10 [2 + 19 \times 2] = 10 [2 + 38] = 10 \times 40 = 400. The sum of the first 20 terms is 400.
• Example 3: Finding the number of terms Q: Which term of the AP 21, 18, 15, ... is -81? A: Here, a = 21, d = 18 - 21 = -3. We are given a_n = -81 and need to find n. Using a_n = a + (n - 1)d, -81 = 21 + (n - 1)(-3) -81 - 21 = -3(n - 1) -102 = -3(n - 1) 34 = n - 1 n = 35. So, -81 is the 35th term.

Exam Tips for Arithmetic Progressions

1. Read Carefully: Distinguish whether the question asks for the n-th term (a_n) or the sum of n terms (S_n). These are the most common sources of error.
2. Verify Common Difference: Always calculate d using at least two pairs of consecutive terms (e.g., a_2 - a_1 and a_3 - a_2) to avoid calculation mistakes, especially when negative numbers are involved.
3. Units and Context: Pay attention to any real-world problem scenarios. Ensure your final answer makes sense in the given context (e.g., 'number of terms' cannot be a fraction or negative).
4. Practice Derivations: While not always asked directly, understanding how the S_n formula is derived (by writing terms forwards and backwards) helps in conceptual clarity and memory retention.
5. Simultaneous Equations: Many problems involve two conditions, leading to two linear equations in a and d. Be proficient in solving such systems to find the unknowns.

Quick Revision Check

• Q: What is the common difference of the AP 0.6, 1.7, 2.8, ...? A: d = 1.7 - 0.6 = 1.1.
• Q: If the first term of an AP is 5 and the common difference is 3, what is the 4th term? A: a_4 = a + (4-1)d = 5 + 3(3) = 5 + 9 = 14.
• Q: Can the common difference of an AP be zero? Give an example. A: Yes, d can be zero. Example: 7, 7, 7, 7, ...
• Q: If a_n = 3n - 2, is this an AP? If so, find its common difference. A: Yes, it is an AP. a_1 = 1, a_2 = 4, a_3 = 7. d = a_2 - a_1 = 4 - 1 = 3.

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