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Mixed Arithmetic & Geometric Series

Introduction

In a Mixed Arithmetic & Geometric Series, the terms follow a combination of two rules - one arithmetic (addition or subtraction by a fixed number) and one geometric (multiplication or division by a fixed ratio). These series test your ability to recognize blended patterns where both additive and multiplicative changes are applied alternately or sequentially.

You may find patterns such as ×2 + 1, ×3 - 2, or alternations like +2, ×2, +3, ×3. Recognizing which operation applies where is the key.

Pattern: Mixed Arithmetic & Geometric Series

Pattern

The key idea: both addition/subtraction and multiplication/division rules appear in the same series - either alternating or combined in each step.

Formula (general idea):
Tₙ = (Tₙ₋₁ × r) + d or Tₙ = (Tₙ₋₁ + a) × r
where r = multiplication factor (ratio) and d or a = addition/subtraction constant.

Common examples include:
• Multiply then add → ×2 + 1
• Add then multiply → +2 × 2
• Alternate between addition and multiplication each step.

Step-by-Step Example

Question

Find the next term in the series: 2, 5, 11, 23, ?

Solution

  1. Step 1: Observe the pattern

    2 → 5 (+3), 5 → 11 (+6), 11 → 23 (+12). Each addition doubles: +3, +6, +12 → next addition should be +24.

  2. Step 2: Apply the rule

    23 + 24 = 47 → Next term = 47.

  3. Step 3: Verify alternate logic

    Alternatively, pattern can be seen as ×2 + 1: 2×2+1=5, 5×2+1=11, 11×2+1=23, 23×2+1=47.

  4. Final Answer:

    47

  5. Quick Check:

    2×2+1=5 → 5×2+1=11 → 11×2+1=23 → 23×2+1=47 ✅

Quick Variations

1. Constant multiplication with constant addition (e.g., ×2 + 3).

2. Alternate addition and multiplication (e.g., +2, ×2, +3, ×3).

3. Multiplication and subtraction (e.g., ×3 - 2).

4. Complex blends where rules repeat every 2 or 3 steps.

Trick to Always Use

  • Check if the change alternates between + and ×.
  • Write down both the additive and multiplicative differences separately.
  • Try combining them - e.g., multiply then add, or add then multiply.
  • Confirm by testing the rule on all previous terms.

Summary

Summary

  • Mixed series combine arithmetic and geometric progressions.
  • Look for alternating or blended (+, ×) patterns.
  • Use both addition and multiplication tests to confirm the logic.
  • Always validate the rule on at least three consecutive terms.

Example to remember:
2, 5, 11, 23 → ×2 + 1 pattern → 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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