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Arithmetic Progression (A.P.) – nth Term

Introduction

Arithmetic Progression (A.P.) aptitude और reasoning tests का एक बहुत ही basic और important pattern है। इसमें numbers की एक sequence होती है जिसमें consecutive terms के बीच का difference हमेशा constant रहता है। किसी भी term (nth term) को जल्दी से निकालना सीखना कई तरह के series-based questions को आसानी से solve करने में मदद करता है।

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

Pattern

nth term का formula: Tₙ = a + (n - 1)d

यहाँ a = first term, d = common difference, और n = term number है।

Step-by-Step Example

Question

निम्न A.P. का 15वाँ term निकालें: 3, 7, 11, 15, …

Solution

  1. Step 1: a और d पहचानें

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

  2. Step 2: nth term का formula लिखें

    Tₙ = a + (n - 1)d

  3. Step 3: Values substitute करके compute करें

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

  4. Final Answer:

    15वाँ term = 59

  5. Quick Check:

    Sequence हर बार 4 से बढ़ रहा है: 3, 7, 11, 15, … → 15वाँ term = 59 ✅

Quick Variations

1. A.P. में किसी term की position निकालना (reverse problem)।

2. nth term formula का use करके missing terms निकालना।

3. Word problems जैसे salary, ages, seats आदि में A.P. logic लगाना।

Trick to Always Use

  • Step 1: हमेशा a और d पहले clearly लिखें।
  • Step 2: Confusion हो तो दो consecutive terms subtract करके d निकालें।
  • Step 3: Formula Tₙ = a + (n - 1)d बहुत ध्यान से apply करें - गलती ज़्यादातर (n - 1) miss करने से होती है।

Summary

Summary

A.P. में:

  • Consecutive terms का difference हमेशा constant होता है।
  • Formula: Tₙ = a + (n - 1)d
  • Sequence का कोई भी term या उसकी position इस formula से निकाली जा सकती है।
  • हमेशा difference pattern से verify करें।

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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