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Arithmetic Progression (A.P.) – nth Term

Introduction

Arithmetic Progression (A.P.) is one of the most fundamental patterns in aptitude and reasoning tests. It represents a sequence of numbers where the difference between consecutive terms remains constant. Understanding how to find any term (nth term) in such a sequence is essential for solving a variety of series-based problems quickly.

Pattern: Arithmetic Progression (A.P.) – nth Term

Pattern

The nth term of an A.P. is given by the formula: Tₙ = a + (n - 1)d

Here, a = first term, d = common difference, and n = term number.

Step-by-Step Example

Question

Find the 15th term of the A.P.: 3, 7, 11, 15, …

Solution

  1. Step 1: Identify a and d:

    First term, a = 3
    Common difference, d = 7 - 3 = 4

  2. Step 2: Use the nth term formula:

    Tₙ = a + (n - 1)d

  3. Step 3: Substitute values and compute:

    T₁₅ = 3 + (15 - 1) × 4 = 3 + 14 × 4 = 3 + 56 = 59

  4. Final Answer:

    The 15th term is 59.

  5. Quick Check:

    Sequence increases by 4 each time: 3, 7, 11, 15, … → 15th term = 59 ✅

Quick Variations

1. Finding which term of the A.P. equals a given value (reverse problem).

2. Using the nth term to find missing terms in between.

3. Applying nth term formula to word problems (ages, salaries, seats, etc.).

Trick to Always Use

  • Step 1: Always write down a and d clearly before substituting.
  • Step 2: If terms look confusing, subtract two consecutive terms to find d.
  • Step 3: Use Tₙ = a + (n - 1)d carefully - most mistakes happen by missing (n - 1).

Summary

Summary

In an Arithmetic Progression (A.P.):

  • The difference between consecutive terms is constant.
  • Formula: Tₙ = a + (n - 1)d.
  • Used to find any term in the sequence or identify its position.
  • Always verify using the common difference pattern.

Practice

(1/5)
1. Find the 10th term of the A.P.: 2, 5, 8, 11, …
easy
A. 26
B. 27
C. 28
D. 29

Solution

  1. Step 1: Identify a and d:

    First term a = 2. Common difference d = 5 - 2 = 3.

  2. Step 2: Use the nth term formula:

    Tₙ = a + (n - 1)d

  3. Step 3: Substitute values and compute:

    T₁₀ = 2 + (10 - 1) × 3 = 2 + 9 × 3 = 2 + 27 = 29

  4. Final Answer:

    The 10th term is 29 → Option D.

  5. Quick Check:

    Add 3 repeatedly: 2, 5, 8, 11, ... (10th term = 29) ✅

Hint: Identify a and d quickly; then apply Tₙ = a + (n-1)d.
Common Mistakes: Using n instead of (n-1) in the formula.
2. Find the 12th term of the A.P.: 4, 9, 14, 19, …
easy
A. 59
B. 64
C. 54
D. 60

Solution

  1. Step 1: Identify a and d:

    First term a = 4. Common difference d = 9 - 4 = 5.

  2. Step 2: Use the nth term formula:

    Tₙ = a + (n - 1)d

  3. Step 3: Substitute values and compute:

    T₁₂ = 4 + (12 - 1) × 5 = 4 + 11 × 5 = 4 + 55 = 59

  4. Final Answer:

    The 12th term is 59 → Option A.

  5. Quick Check:

    Sequence increases by 5: 4,9,14,19,... → 12th term = 59 ✅

Hint: Subtract consecutive terms to get d, then multiply d by (n-1).
Common Mistakes: Incorrect arithmetic when multiplying (n-1) by d.
3. The 1st term of an A.P. is 6 and the 8th term is 34. Find the common difference.
easy
A. 3
B. 4
C. 5
D. 6

Solution

  1. Step 1: Use the nth term formula for T₈:

    T₈ = a + (8 - 1)d ⇒ 34 = 6 + 7d.

  2. Step 2: Solve for d:

    34 - 6 = 7d ⇒ 28 = 7d ⇒ d = 28 ÷ 7 = 4.

  3. Final Answer:

    Common difference d = 4 → Option B.

  4. Quick Check:

    Terms: 6,10,14,18,22,26,30,34 ✅

Hint: Use d = (Tₙ - a) / (n - 1).
Common Mistakes: Forgetting to subtract the first term before dividing by (n-1).
4. Which term of the A.P. 5, 11, 17, 23, … is 65?
medium
A. 9th
B. 10th
C. 11th
D. 12th

Solution

  1. Step 1: Identify a and d:

    First term a = 5. Common difference d = 11 - 5 = 6.

  2. Step 2: Set up Tₙ = given value:

    65 = 5 + (n - 1)×6.

  3. Step 3: Solve for n:

    65 - 5 = 6(n - 1) ⇒ 60 = 6(n - 1) ⇒ n - 1 = 10 ⇒ n = 11.

  4. Final Answer:

    65 is the 11th term → Option C.

  5. Quick Check:

    11th term = 5 + 10×6 = 5 + 60 = 65 ✅

Hint: Rearrange n = [(Tₙ - a)/d] + 1.
Common Mistakes: Forgetting to add 1 after division.
5. In an A.P., the 3rd term is 12 and the 7th term is 24. Find the first term.
medium
A. 6
B. 5
C. 4
D. 3

Solution

  1. Step 1: Write equations for T₃ and T₇:

    T₃ = a + 2d = 12, and T₇ = a + 6d = 24.

  2. Step 2: Subtract to find d:

    (a + 6d) - (a + 2d) = 24 - 12 ⇒ 4d = 12 ⇒ d = 3.

  3. Step 3: Substitute d to find a:

    a + 2×3 = 12 ⇒ a + 6 = 12 ⇒ a = 6.

  4. Final Answer:

    First term a = 6 → Option A.

  5. Quick Check:

    Sequence: 6,9,12,15,18,21,24 ✅

Hint: Form two equations for the given terms, subtract to find d, then back-substitute to get a.
Common Mistakes: Arithmetic errors while subtracting equations or when substituting d.

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