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Mixed Series (Combination of A.P. and G.P.)

Introduction

कुछ sequences दो या अधिक प्रकार के patterns मिलाकर बनते हैं - आम तौर पर एक Arithmetic Progression (A.P.) और एक Geometric Progression (G.P.). इन्हें Mixed Series कहा जाता है। यह पहचानना कि किस हिस्से पर कौन सा नियम लागू है (additive बनाम multiplicative) इनको जल्दी और सही हल करने की कुंजी है।

यह pattern इसलिए महत्वपूर्ण है क्योंकि कई reasoning questions A.P. और G.P. दोनों को mix करके आपकी observation और analytical skills टेस्ट करते हैं। ऐसी series की dual प्रकृति पहचानने से समय बचता है और exam में confusion कम होता है।

Pattern: Mixed Series (Combination of A.P. and G.P.)

Pattern

एक Mixed Series में या तो alternate terms या हर term के components अलग नियम follow करते हैं - एक arithmetic (constant difference) और एक geometric (constant ratio)।

Common types में शामिल हैं:

  • Type 1 - Alternating pattern: Odd terms एक A.P. बनाते हैं, even terms एक G.P. बनाते हैं (या इसके विपरीत)।
  • Type 2 - Additive + multiplicative combination: हर term किसी constant को जोड़ने के बाद किसी fixed number से multiply करके मिलता है।
  • Type 3 - Dual rule progression: Term alternating तरीके से fixed addition और फिर multiplication से बढ़ता है।

इसे हल करने के लिए: series को दो sequences में विभाजित करें - एक odd positions के लिए और एक even positions के लिए। फिर चेक करें कि क्या हर हिस्सा A.P. या G.P. के नियमों का पालन करता है।

Step-by-Step Example

Question

इस series का अगला term निकालें: 2, 4, 8, 10, 20, 22, 44, …

Solution

  1. Step 1: Odd और Even positions अलग करें

    Odd terms: 2, 8, 20, 44
    Even terms: 4, 10, 22

  2. Step 2: Odd-term pattern चेक करें

    Odd terms (2, 8, 20, 44) - ratio देखें: 8/2=4, 20/8=2.5, 44/20=2.2 → यह G.P. नहीं है।
    Difference देखें: 8-2=6, 20-8=12, 44-20=24 → differences हर बार दोगुने हो रहे हैं (differences में वृद्धि)।

  3. Step 3: Even-term pattern चेक करें

    Even terms (4, 10, 22) - 10-4=6, 22-10=12 → यहाँ भी differences दोगुना होते दिख रहे हैं।

  4. Step 4: Mixed rule पहचानें

    Series alternate कर रहा है और दोनों halves में ऐसा additive pattern दिखता है जहाँ gap हर बार दोगुना हो जाता है। दोनों हिस्से एक ही doubling-difference नियम को mirror करते हैं।

  5. Step 5: Next term का prediction करें

    Even-term sequence का next difference = 12 × 2 = 24 → next even term = 22 + 24 = 46.

  6. Final Answer:

    Next term = 46.

  7. Quick Check:

    Odd और even दोनों subseries doubling-difference नियम का पालन करते हैं → consistent ✅

Quick Variations

1. Odd-Even separation: एक half A.P. बनाता है, दूसरा half G.P. बनाता है।

2. हर term = (previous term × fixed number) + constant।

3. Alternate addition और multiplication, उदाहरण: ×2, +3, ×2, +3, …

4. एक half linear growth दिखाता है, दूसरा exponential।

5. एक ही formula में mix: Tₙ = 2n × 3ⁿ या Tₙ = n² × 2ⁿ जैसे रूप हो सकते हैं।

Trick to Always Use

  • Step 1 → Series को odd और even positions में अलग करें।
  • Step 2 → हर हिस्से को A.P. (constant difference) या G.P. (constant ratio) के लिए टेस्ट करें।
  • Step 3 → अगर कोई भी ठीक से फिट नहीं होता तो alternate add-multiply patterns देखें।
  • Step 4 → sub-series logic का उपयोग करके प्रत्येक समूह का अगला term निकालें।
  • Step 5 → दोनों हिस्सों को वापस मिलाकर consistency verify करें।

Summary

Summary

  • Mixed Series additive (A.P.) और multiplicative (G.P.) patterns को मिलाते हैं।
  • Odd और even terms को अलग करना अक्सर नियम जल्दी खोल देता है।
  • Common mixed forms में alternate add-multiply या स्वतंत्र नियमों वाले sub-series शामिल होते हैं।
  • हमेशा differences और ratios दोनों को calculate करके rule confirm करें।
  • Same sub-rule आगे apply करके missing या next terms निकालें।

Practice

(1/5)
1. Find the next term in the series: 1, 2, 3, 6, 5, 18, __
easy
A. 7
B. 54
C. 20
D. 9

Solution

  1. Step 1: Split odd and even positions

    Odd-position terms: 1 (pos1), 3 (pos3), 5 (pos5) → sequence: 1, 3, 5.

    Even-position terms: 2 (pos2), 6 (pos4), 18 (pos6) → sequence: 2, 6, 18.

  2. Step 2: Identify rules for each subsequence

    Odd subsequence 1,3,5 is an A.P. with common difference +2 (1 → 3 → 5).

    Even subsequence 2,6,18 is a G.P. with common ratio ×3 (2×3=6, 6×3=18).

  3. Step 3: Find the next term

    The next overall term is position 7 (odd). Continue the odd A.P.: next odd-term = 5 + 2 = 7.

  4. Final Answer:

    Next term = 7 → Option A.

  5. Quick Check:

    Odd positions now: 1,3,5,7 (A.P. with +2). Even positions still: 2,6,18 (G.P. with ×3). Subseries rules hold ✅

Hint: Split into odd/even positions - handle each with A.P. or G.P. separately.
Common Mistakes: Treating whole sequence as one progression instead of two interleaved subsequences.
2. Find the next term in the series: 4, 2, 7, 6, 10, 18, __
easy
A. 13
B. 36
C. 27
D. 20

Solution

  1. Step 1: Separate odd and even terms

    Odd terms (positions 1,3,5): 4, 7, 10 → these look like an A.P.

    Even terms (positions 2,4,6): 2, 6, 18 → these look like a G.P.

  2. Step 2: Determine subsequence rules

    Odd subsequence 4,7,10 is an A.P. with d = 3 (4→7→10).

    Even subsequence 2,6,18 is a G.P. with r = 3 (2×3=6, 6×3=18).

  3. Step 3: Compute next term

    Next overall term is position 7 (odd). Continue odd A.P.: next odd-term = 10 + 3 = 13.

  4. Final Answer:

    Next term = 13 → Option A.

  5. Quick Check:

    Odd subsequence becomes 4,7,10,13 (A.P. with +3). Even subsequence remains 2,6,18 (G.P. with ×3) ✅

Hint: When one subsequence is A.P. and the other G.P., handle each separately and recombine.
Common Mistakes: Applying the ratio to the A.P. subsequence or vice versa.
3. Find the next term in the series: 5, 10, 15, 30, 45, 90, __
easy
A. 120
B. 135
C. 150
D. 90

Solution

  1. Step 1: Look at odd and even positions

    Odd positions: 5 (pos1), 15 (pos3), 45 (pos5) → 5, 15, 45.

    Even positions: 10 (pos2), 30 (pos4), 90 (pos6) → 10, 30, 90.

  2. Step 2: Identify rules

    Both subsequences are G.P. with ratio ×3: 5→15→45 and 10→30→90.

  3. Step 3: Find next term

    Next overall term is position 7 (odd). Continue odd G.P.: next odd-term = 45 × 3 = 135.

  4. Final Answer:

    Next term = 135 → Option B.

  5. Quick Check:

    Odd subsequence 5,15,45,135 (×3). Even subsequence 10,30,90 (×3) - consistent ✅

Hint: If both subsequences multiply by same ratio, just extend the appropriate subsequence for the next index.
Common Mistakes: Trying to apply a single rule to the whole series rather than to each subsequence.
4. Find the next term in the series: 4, 3, 9, 6, 14, 12, 19, __
medium
A. 20
B. 22
C. 24
D. 26

Solution

  1. Step 1: Split into subsequences by parity of position

    Odd positions: 4 (pos1), 9 (pos3), 14 (pos5), 19 (pos7) → this is an A.P.

    Even positions: 3 (pos2), 6 (pos4), 12 (pos6) → this is a G.P.

  2. Step 2: Determine each rule

    Odd subsequence 4,9,14,19 is an A.P. with common difference +5.

    Even subsequence 3,6,12 is a G.P. with ratio ×2 (3→6→12).

  3. Step 3: Compute the next term

    Next overall term is position 8 (even). Continue even G.P.: next even-term = 12 × 2 = 24.

  4. Final Answer:

    Next term = 24 → Option C.

  5. Quick Check:

    Odd subsequence remains A.P. (+5). Even subsequence becomes 3,6,12,24 (×2) - both patterns hold ✅

Hint: Label positions odd/even and test for A.P. or G.P. separately before choosing the next term.
Common Mistakes: Confusing which subsequence controls the next index (odd vs even).
5. Find the next term in the series: 1, 2, 5, 10, 13, 26, __
medium
A. 28
B. 26
C. 32
D. 29

Solution

  1. Step 1: Observe alternating operations

    Check the pattern term-to-term: 1 → 2 (×2), 2 → 5 (+3), 5 → 10 (×2), 10 → 13 (+3), 13 → 26 (×2).

  2. Step 2: Identify the rule

    The sequence alternates: multiply by 2, then add 3, repeating: ×2, +3, ×2, +3, ×2, …

  3. Step 3: Apply the next operation

    Last operation was ×2 (13→26), so next is +3: 26 + 3 = 29.

  4. Final Answer:

    Next term = 29 → Option D.

  5. Quick Check:

    Sequence built as: 1, (1×2)=2, (2+3)=5, (5×2)=10, (10+3)=13, (13×2)=26, (26+3)=29 ✅

Hint: When operations alternate, write the operation sequence (×, +, ×, +, …) and apply to the last term.
Common Mistakes: Applying the wrong next operation (e.g., multiplying again instead of adding).

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