Arithmetic Progressions

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🔢 What is an Arithmetic Progression?

A sequence of numbers where the difference between consecutive terms is constant. That constant difference is called the Common Difference (d).

2

5

8

11

14

…

Common Difference d = a₂ − a₁ = a₃ − a₂ = … = constant

Here: 5 − 2 = 3, 8 − 5 = 3, 11 − 8 = 3 → d = 3
First term a = 2, Common difference d = 3
If d > 0 → Increasing AP | If d < 0 → Decreasing AP

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📌 The n^th Term Formula

To find any term without listing all terms, we use the nth term formula:

1

Start with first term: a₁ = a

The 1st term is just a.

2

Add d once to get a₂

a₂ = a + d

3

Add d twice to get a₃

a₃ = a + 2d

4

Pattern: aₙ = a + (n−1)d

For the nth term, we add d exactly (n−1) times.

          🌟 nth Term Formula
          aₙ = a + (n − 1) × d
        

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📊 Visualising AP: a=3, d=4

Watch each term grow by a constant amount (d=4). The bars increase uniformly — that's the beauty of AP!

3

a₁

7

a₂

11

a₃

15

a₄

19

a₅

23

a₆

a = 3, d = 4 → 3, 7, 11, 15, 19, 23, …

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➕ Sum of n Terms (Sₙ)

Gauss's trick — pair the first and last terms. Each pair sums to the same value!

1

Write Sₙ forward and backward

Sₙ = a + (a+d) + (a+2d) + … + l where l = last term

2

Add both rows

2Sₙ = n × (a + l) → each pair sums to (a + l)

3

Substitute l = a + (n−1)d

2Sₙ = n × (2a + (n−1)d)

4

Divide both sides by 2

Sₙ = n/2 × (2a + (n−1)d) ✓

          🌟 Sum Formula
          Sₙ = n/2 × [2a + (n−1)d]   OR   Sₙ = n/2 × (a + l)
        

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✏️ Solved Example

Q: Find the 10th term and sum of first 10 terms of AP: 5, 8, 11, 14, …

1

Identify a and d

a = 5, d = 8 − 5 = 3

2

Find a₁₀ using aₙ = a + (n−1)d

a₁₀ = 5 + (10−1) × 3 = 5 + 27 = 32

3

Find S₁₀ using Sₙ = n/2(2a + (n−1)d)

S₁₀ = 10/2 × (2×5 + 9×3) = 5 × (10 + 27) = 5 × 37 = 185

a₁₀ = 32 S₁₀ = 185

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🧠 Quick Quiz

Q: What is the 7th term of AP: 2, 5, 8, 11, … ?

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