An Arithmetic Progression (AP) is simply a sequence of numbers with a consistent pattern: the same number is added or subtracted each time to get the next number. This topic is formula-based, so memorizing the key formulas and knowing when to use them is the secret to success.

1. Core Concepts & Key Terminology

• What is an AP? A list of numbers where the "step" between them is constant.

  • Example 1 (Increasing): 4, 10, 16, 22, ... (The step is adding 6)

  • Example 2 (Decreasing): 50, 45, 40, 35, ... (The step is subtracting 5)

• Key Terms (The variables for your formulas):

  • a (First Term): The number that starts the sequence.

    • In Example 1, a = 4.

  • d (Common Difference): The "step" you take each time. Crucially, this can be negative.

    • d = (Second Term) - (First Term)

    • In Example 1, d = 10 - 4 = 6.

    • In Example 2, d = 45 - 50 = -5.

  • n (Number of Terms): The position of a term in the sequence (e.g., 5th term, 20th term).

  • Tₙ (The nth Term): The value of the term at position n. T₃ in Example 1 is 16.

  • Sₙ (Sum of the first n terms): The total you get when you add up the first n terms of the sequence.

2. The Three Essential Formulas (MEMORIZE THESE)

All exam questions will use these.

1. The Term Formula (to find the value of any term): Tₙ = a + (n-1)d

2. The Sum Formula (when you know a and d): Sₙ = n/2 [2a + (n-1)d]

3. The Sum Formula (when you know the first term a and last term l): Sₙ = n/2 (a + l)

3. How to Solve Exam Questions (Step-by-Step)

Exam questions fall into a few common types. Here’s how to handle them.

Type 1: Find a Specific Term

• Question: "Find the 30th term of the sequence 7, 11, 15, ..."

• Steps:

  1. Identify: a = 7, d = 11 - 7 = 4, n = 30.

  2. Formula: Use the Term Formula: Tₙ = a + (n-1)d.

  3. Solve: T₃₀ = 7 + (30-1) × 4 = 7 + (29 × 4) = 7 + 116 = 123.

Type 2: Find the Sum of a Number of Terms

• Question: "Find the sum of the first 20 terms of the sequence 50, 45, 40, ..."

• Steps:

  1. Identify: a = 50, d = 45 - 50 = -5, n = 20.

  2. Formula: Use the Sum Formula: Sₙ = n/2 [2a + (n-1)d].

  3. Solve: S₂₀ = 20/2 [2(50) + (20-1) × -5] = 10 [100 + 19 × -5] = 10 [100 - 95] = 10 × 5 = 50.

Type 3: Find a and d using Simultaneous Equations

• Clue: The question gives you two pieces of information, like "The 4th term is 11 and the 9th term is 26."

• Steps:

  1. Set up two equations using the Term Formula:

     • T₄ = a + (4-1)d => 11 = a + 3d (Equation 1)

     • T₉ = a + (9-1)d => 26 = a + 8d (Equation 2)

  2. Solve the simultaneous equations: (Subtract Eq 1 from Eq 2)

     • (a + 8d) - (a + 3d) = 26 - 11

     • 5d = 15 => d = 3.

  3. Substitute d=3 back into Equation 1 to find a:

     • 11 = a + 3(3) => 11 = a + 9 => a = 2.

Type 4: Find 'n' (how many terms)

• Question: "How many terms of the AP 2, 6, 10, ... must be added to get a sum of 200?"

• Steps:

  1. Identify: a = 2, d = 4, Sₙ = 200. You need to find n.

  2. Formula: Use the Sum Formula: Sₙ = n/2 [2a + (n-1)d].

  3. Substitute and simplify into a quadratic equation:

     • 200 = n/2 [2(2) + (n-1) × 4]

     • 200 = n/2 [4 + 4n - 4]

     • 200 = n/2 [4n]

     • 200 = 2n²

  4. Solve the quadratic equation:

     • n² = 100

     • n = 10 (since n cannot be negative).

     • Answer: 10 terms.

4. Important Points to Remember

• d can be negative! For a decreasing sequence, d will be a negative number. This is a very common place to make a mistake.

• Tₙ vs. Sₙ: Read the question carefully. "Find the 10th term" means find T₁₀. "Find the sum of the first 10 terms" means find S₁₀.

• Word Problems: The first step is to identify a and d from the story. For example, if someone saves Rs. 50 in the first week and increases the savings by Rs. 10 each week, then a=50 and d=10.