Class 10 Maths Chapter 5: Arithmetic Progressions Concept Infographic

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Chapter 5: Arithmetic Progressions

Decoding numerical patterns, common differences, and sum of series[cite: 6].

📘 What is an AP?

The Core Definition

An AP is a list of numbers where each term is obtained by adding a fixed number to the preceding term (except the first term)[cite: 6].

Common Difference (d)
The fixed number is called 'd'. It can be positive, negative, or zero[cite: 6]!

⚙️ The Master Formulas

a_n = a + (n - 1)d

General Term: Finds the n^th term of the AP[cite: 6].

S_n =

n2

[2a + (n - 1)d]

Sum of First n Terms: Use when 'd' is known[cite: 6].

S_n =

n2

(a + l)

Short Sum Formula: Use when the last term (l) is known[cite: 6].

📙 Crucial Properties

General Form

a, a+d, a+2d, a+3d, ...[cite: 6]

Finite vs Infinite
A finite AP has a last term (l). An infinite AP does not have a last term[cite: 6].

Sum of Positive Integers
S_n =

n(n + 1)2

[cite: 6]

⭐ Board Exam Super-Hacks

The "Hidden Term" Hack

The n^th term of an AP is the difference between the sum to first n terms and the sum to first (n - 1) terms[cite: 6]:
a_n = S_n - S_n-1[cite: 6]

Arithmetic Mean Rule

If three numbers a, b, c are in AP, then b = (a + c) / 2[cite: 6]. Here, b is called the Arithmetic Mean of a and c[cite: 6].

⚠️ Common Exam Traps

Trap 1: Calculating 'd' Backwards

To find 'd', always subtract the preceding term from the succeeding term: d = a₂ - a₁[cite: 6]. Do NOT subtract the smaller number from the larger one if the sequence is decreasing!

Trap 2: Fractional 'n' Values

When checking if a number belongs to an AP, 'n' represents the position. Therefore, n MUST be a positive integer[cite: 6]. If you get n = 151/6 or n = -5, that term does not exist in the AP.

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