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Missing Term (Formula-Based)

Introduction

एक Missing Term (Formula-Based) श्रृंखला में, हर पद एक स्पष्ट गणितीय formula या नियम से बनाया जाता है जो term index (n), पिछले पदों या ऑपरेशनों के संयोजन (powers, factorial, multiplication/addition patterns) पर आधारित हो सकता है।

यह पैटर्न महत्वपूर्ण है क्योंकि कई exam-style प्रश्नों में एक छोटा सा कॉम्पैक्ट नियम छिपा होता है (उदाहरण: Tn = n² + 1 या Tn = Tn-1 × n) और आपसे गायब पद को उस formula का उपयोग करके निकालने के लिए कहा जाता है।

Pattern: Missing Term (Formula-Based)

Pattern

मुख्य विचार: ऐसा गणितीय formula पहचानें जो हर पद बनाता है, और उसी का उपयोग करके missing value निकालें।

Formula (useful forms you may encounter):
Tn = a·n + b - linear formula (arithmetic progression).
Tn = a·r^(n-1) - geometric progression.
Tn = an² + bn + c - quadratic formula (second differences देखें)।
Tn = Tn-1 · f(n) या Tn = Tn-1 + g(n) - recursive/formula-based rules.
Index-based होने पर, हमेशा पद को n (1,2,3,...) के फ़ंक्शन के रूप में लिखने की कोशिश करें।

Step-by-Step Example

Question

Missing term खोजें: 2, 6, 12, 20, ?, 42.

Solution

  1. Step 1: Positions देखें और formulas आज़माएँ

    पदों के नीचे index लिखें n = 1,2,3,4,5,6 → 2, 6, 12, 20, ?, 42.
    सरल index-based formulas जैसे n², n(n+1), n²+1 आदि परीक्षण करें।

  2. Step 2: संभावित formula n(n+1) टेस्ट करें

    n=1..6 के लिए n(n+1) निकालें: 1·2=2, 2·3=6, 3·4=12, 4·5=20, 5·6=30, 6·7=42.
    ये ज्ञात पदों से बिल्कुल मेल खाते हैं।

  3. Step 3: Missing position पर formula लागू करें

    n = 5 → 5·6 = 30.

  4. Final Answer:

    30

  5. Quick Check:

    n(n+1) से श्रृंखला: 2, 6, 12, 20, 30, 42 → blank के दोनों ओर सही बैठता है ✅

Quick Variations

1. Index-based polynomials: Tn = an² + bn + c - second differences जाँचें।

2. Factorial-like growth: Tn = n! या n! ± k - बड़े जंप; factorial के संकेत देखें।

3. Recursive formulas: Tn = Tn-1 + p(n) या × q(n) - consecutive terms का संबंध जाँचें।

4. Mixed index + previous-term rules: Tn = n·Tn-1 या Tn = Tn-1 + n² आदि।

Trick to Always Use

  • Step 1 → Index substitution करें: श्रृंखला के नीचे (1,2,3,...) लिखें।
  • Step 2 → सरल forms क्रम में टेस्ट करें: linear (an+b), n(n+1), squares/cubes, factorials।
  • Step 3 → अगर differences constant न हों, first और second differences निकालकर quadratic पैटर्न पहचानें।
  • Step 4 → Recursive rule के लिए ratios या term/previous-term relation देखें।

Summary

Summary

  • पदों को उनके index (n) के फ़ंक्शन के रूप में व्यक्त करने की कोशिश करें।
  • Linear या quadratic rule पहचानने के लिए differences का उपयोग करें।
  • Index-based formula न फिट होने पर recursive rules जाँचें।
  • उत्तर को formula में दोबारा रखकर या पड़ोसी पदों से मिलाकर quick-check करें।

याद रखें: n(n+1) formula 2,6,12,20,30,... जैसी बहुत common series बनाता है।

Practice

(1/5)
1. Find the missing term: 7, 10, 13, ?, 19
easy
A. 16
B. 15
C. 18
D. 14

Solution

  1. Step 1: Observe differences

    10 - 7 = 3, 13 - 10 = 3, 19 - ? should also fit the same gap.

  2. Step 2: Identify the formula

    Constant difference → arithmetic progression with common difference d = 3 (linear formula Tₙ = a₁ + (n-1)d).

  3. Step 3: Apply the rule

    Next term after 13 = 13 + 3 = 16.

  4. Final Answer:

    16 → Option A

  5. Quick Check:

    Sequence with 16: 7,10,13,16,19 → differences all = 3 ✅
Hint: Check first differences; constant difference → arithmetic progression.
Common Mistakes: Assuming a multiplicative rule when differences are constant.
2. Find the missing term: 3, 6, 12, ?, 48
easy
A. 12
B. 24
C. 18
D. 30

Solution

  1. Step 1: Check ratios

    6 ÷ 3 = 2, 12 ÷ 6 = 2, 48 ÷ ? should also equal 2 if geometric.

  2. Step 2: Identify the formula

    Common ratio r = 2 → geometric progression Tₙ = a·r^(n-1).

  3. Step 3: Apply the rule

    Next term after 12 = 12 × 2 = 24.

  4. Final Answer:

    24 → Option B

  5. Quick Check:

    Sequence: 3,6,12,24,48 → each term ×2 ✅
Hint: Test multiplicative ratio (consecutive division) for geometric sequences.
Common Mistakes: Mixing up additive and multiplicative rules.
3. Find the missing term: 2, 5, 10, 17, ?
easy
A. 25
B. 24
C. 26
D. 27

Solution

  1. Step 1: Try index-based formula

    Test n² + 1 for n = 1,2,3,4,...: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17.

  2. Step 2: Identify the formula

    So Tₙ = n² + 1 fits the sequence.

  3. Step 3: Apply the rule

    For n = 5 → 5² + 1 = 26.

  4. Final Answer:

    26 → Option C

  5. Quick Check:

    Values: 1²+1,2²+1,3²+1,4²+1,5²+1 → 2,5,10,17,26 ✅
Hint: Try n² ± k or n³ ± k when differences grow non-linearly.
Common Mistakes: Using first differences only - index formulas often use n explicitly.
4. Find the missing term: 6, 11, 18, 27, ?
medium
A. 36
B. 35
C. 40
D. 38

Solution

  1. Step 1: Compute first differences

    11 - 6 = 5, 18 - 11 = 7, 27 - 18 = 9 → D₁ = 5,7,9.

  2. Step 2: Check second differences

    7 - 5 = 2, 9 - 7 = 2 → constant second difference = +2, indicating a quadratic formula Tₙ = an² + bn + c.

  3. Step 3: Extend the difference sequence

    Next first difference = 9 + 2 = 11 → Next term = 27 + 11 = 38.

  4. Final Answer:

    38 → Option D

  5. Quick Check:

    First diffs 5,7,9,11 → terms 6,11,18,27,38 ✅
Hint: If second differences are constant, use quadratic extension (add same increment to first differences).
Common Mistakes: Forcing arithmetic progression when second differences reveal quadratic behavior.
5. Find the missing term: 2, 6, 24, ?, 720
medium
A. 120
B. 144
C. 96
D. 480

Solution

  1. Step 1: Recognise rapid growth

    Sequence grows very fast: 2 → 6 → 24 → ? → 720. Test factorials: 2!=2, 3!=6, 4!=24, 5!=120, 6!=720.

  2. Step 2: Identify the formula

    Terms follow n! starting at n=2 (i.e., T₁=2!, T₂=3!, ...).

  3. Step 3: Apply the rule

    Missing term corresponds to 5! = 120.

  4. Final Answer:

    120 → Option A

  5. Quick Check:

    Factorials: 2!,3!,4!,5!,6! → 2,6,24,120,720 ✅
Hint: For very fast growth, test factorials or powers (n! or n^k).
Common Mistakes: Assuming geometric progression when factorial growth is present.

Mock Test

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