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Mixed Arithmetic & Geometric Series

Introduction

Mixed Arithmetic & Geometric Series में terms दो अलग-अलग rules को मिलाकर चलते हैं - एक arithmetic (fixed जोड़ या घटाव) और दूसरा geometric (fixed multiplication या division)। ऐसे series आपकी ability को test करते हैं कि आप additive और multiplicative बदलावों को एक साथ या बारी-बारी से पहचान सकें।

अक्सर patterns ×2 + 1, ×3 - 2, या alternating rules जैसे +2, ×2, +3, ×3 देखने को मिलते हैं। पैटर्न कब बदल रहा है, यह पहचानना सबसे महत्वपूर्ण है।

Pattern: Mixed Arithmetic & Geometric Series

Pattern

मुख्य आइडिया: एक ही series में addition/subtraction और multiplication/division दोनों rules मिलकर चलते हैं - या तो हर step पर mix होते हैं या alternating pattern में आते हैं।

General Formula (concept):
Tₙ = (Tₙ₋₁ × r) + d या Tₙ = (Tₙ₋₁ + a) × r
जहाँ r = multiplication ratio और d / a = addition/subtraction constant है।

Common examples:
• Multiply then add → ×2 + 1
• Add then multiply → +2 × 2
• Pure alternation → हर step पर कभी +, कभी ×

Step-by-Step Example

Question

Series का अगला term बताइए: 2, 5, 11, 23, ?

Solution

  1. Step 1: Pattern को observe करें

    2 → 5 (+3), 5 → 11 (+6), 11 → 23 (+12). Additions double हो रहे हैं: +3, +6, +12 → अगले step में +24 लगेगा।

  2. Step 2: Rule लागू करें

    23 + 24 = 47

  3. Step 3: Alternate logic से verify करें

    इसे ×2 + 1 pattern भी माना जा सकता है: 2×2+1=5 → 5×2+1=11 → 11×2+1=23 → 23×2+1=47

  4. Final Answer:

    47

  5. Quick Check:

    Pattern consistent: ×2 + 1 हर step पर काम करता है ✅

Quick Variations

1. Constant multiplication + constant addition (जैसे: ×2 + 3)।

2. Alternating rules (जैसे: +2, ×2, +3, ×3)।

3. Multiply + subtract (जैसे: ×3 - 2)।

4. 2 या 3 steps के बाद repeat होने वाले complex blended rules।

Trick to Always Use

  • जाँचें कि series में + और × का alternation है क्या।
  • Additive और multiplicative differences को अलग-अलग लिखें।
  • दोनों को combine करके देखें-कभी multiply then add, कभी add then multiply।
  • Rule को कम-से-कम 3 consecutive terms पर verify जरूर करें।

Summary

Summary

  • Mixed series में arithmetic और geometric दोनों concepts शामिल होते हैं।
  • Alternate या blended (+, ×) rules को पहचानना जरूरी है।
  • Rule की पुष्टि additive और multiplicative tests दोनों से करें।
  • हमेशा पूरा pattern कम-से-कम तीन steps पर fit होना चाहिए।

याद रखने वाला example:
2, 5, 11, 23 → ×2 + 1 → Next = 47

Practice

(1/5)
1. Find the next term in the series: 2, 6, 14, 30, ?
easy
A. 46
B. 54
C. 62
D. 64

Solution

  1. Step 1: Identify the rule

    The sequence follows the rule: each term = previous term × 2 + 2.

  2. Step 2: Verify the pattern

    2×2+2=6, 6×2+2=14, 14×2+2=30 → pattern holds.

  3. Step 3: Apply the rule

    Next term = 30×2 + 2 = 62.

  4. Final Answer:

    62 → Option C

  5. Quick Check:

    Sequence: 2,6,14,30,62 (each = previous ×2 + 2) ✅
Hint: Test ×2 + constant when numbers roughly double each step.
Common Mistakes: Missing the additive constant after multiplication.
2. Find the missing number: 3, 6, 13, 28, ?
easy
A. 57
B. 58
C. 59
D. 60

Solution

  1. Step 1: Observe pattern

    Each term = previous term × 2 + increment increasing by +1.

  2. Step 2: Verify

    3×2+0=6, 6×2+1=13, 13×2+2=28 → pattern confirmed.

  3. Step 3: Apply the rule

    Next = 28×2+3=56+3=59.

  4. Final Answer:

    59 → Option C

  5. Quick Check:

    3,6,13,28,59 ✅
Hint: Identify both the multiplication and growing additive pattern together.
Common Mistakes: Using only the ×2 rule without noticing the increasing addition.
3. Find the next term in the series: 4, 9, 19, 40, ?
medium
A. 81
B. 82
C. 83
D. 84

Solution

  1. Step 1: Observe successive operations

    The operations follow a mixed pattern: 4×2+1=9, 9×2+1=19, 19×2+2=40. The additive increments are +1, +1, +2.

  2. Step 2: Predict next additive increment

    The additive sequence increases slowly: 1, 1, 2 → next increment = 3.

  3. Step 3: Compute next term

    Next term = 40×2 + 3 = 80 + 3 = 83.

  4. Final Answer:

    83 → Option C

  5. Quick Check:

    Operations follow ×2 + (1,1,2,3) → sequence becomes 4, 9, 19, 40, 83.
Hint: When ×2 is consistent but results vary slightly, check if the additive constant grows slowly (e.g., +1, +1, +2, +3).
Common Mistakes: Assuming pure doubling or relying only on first differences instead of checking blended × and + patterns.
4. Find the missing number in the series: 5, 10, 21, 44, ?
medium
A. 91
B. 90
C. 92
D. 93

Solution

  1. Step 1: Observe the rule

    The sequence follows the rule: each term = previous term × 2 + k, where k increases by 1 each step starting at 0.

  2. Step 2: Verify

    5×2+0=10, 10×2+1=21, 21×2+2=44 → additive constants are 0,1,2 respectively.

  3. Step 3: Apply the next increment

    Next additive constant = 3 → Next term = 44×2 + 3 = 88 + 3 = 91.

  4. Final Answer:

    91 → Option A

  5. Quick Check:

    Sequence check: 5,10,21,44,91 (×2 + 0, then +1, then +2, then +3) ✅
Hint: When additive part grows by 1, list the additive constants and extend them.
Common Mistakes: Assuming a fixed additive constant for all steps.
5. Find the next term in the series: 6, 14, 30, 62, ?
medium
A. 126
B. 128
C. 130
D. 132

Solution

  1. Step 1: Recognize pattern

    The rule is: each term = previous ×2 + 2.

  2. Step 2: Verify

    6×2+2=14, 14×2+2=30, 30×2+2=62 → pattern confirmed.

  3. Step 3: Apply rule

    Next = 62 × 2 + 2 = 124 + 2 = 126.

  4. Final Answer:

    126 → Option A

  5. Quick Check:

    Each step doubles then adds 2 → consistent mixed pattern.
Hint: When values nearly double, test ×2 followed by a small addition.
Common Mistakes: Using ×2+1 (wrong) or pure doubling (also wrong).

Mock Test

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