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

Introduction

Some sequences combine two or more types of patterns - typically an Arithmetic Progression (A.P.) and a Geometric Progression (G.P.). These are called Mixed Series. Recognizing which rule governs which part (additive vs. multiplicative) is key to solving them efficiently.

This pattern is important because many reasoning questions mix A.P. and G.P. to test both your observation and analytical skills. Identifying the dual nature of such series saves time and avoids confusion during exams.

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

Pattern

In a Mixed Series, either alternate terms or components of each term follow different rules - one arithmetic (constant difference) and one geometric (constant ratio).

Common types include:

  • Type 1 - Alternating pattern: Odd terms form an A.P., even terms form a G.P. (or vice versa).
  • Type 2 - Additive + multiplicative combination: Each term is obtained by adding a constant and then multiplying by a fixed number.
  • Type 3 - Dual rule progression: Term increases alternately by fixed addition and multiplication.

To solve: Separate the series into two sequences - one for odd positions and one for even. Check if each follows A.P. or G.P. rules.

Step-by-Step Example

Question

Find the next term in the series: 2, 4, 8, 10, 20, 22, 44, …

Solution

  1. Step 1: Separate odd and even positions

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

  2. Step 2: Check the odd-term pattern

    Odd terms (2, 8, 20, 44) - ratio pattern? 8/2=4, 20/8=2.5, 44/20=2.2 → not G.P.
    Difference pattern? 8-2=6, 20-8=12, 44-20=24 → differences double each time (A.P. in differences).

  3. Step 3: Check the even-term pattern

    Even terms (4, 10, 22) - 10-4=6, 22-10=12 → differences double as well.

  4. Step 4: Identify mixed rule

    The series alternates between A.P. and G.P.-like growth where the gap doubles. Both halves mirror the same additive pattern doubling each time.

  5. Step 5: Predict the next term

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

  6. Final Answer:

    Next term = 46.

  7. Quick Check:

    Both odd and even subseries follow doubling-difference rule → consistent ✅

Quick Variations

1. Odd-Even separation: one forms A.P., the other forms G.P.

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

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

4. One half uses linear growth, the other exponential.

5. Mixed within same formula: Tₙ = 2n × 3ⁿ or Tₙ = n² × 2ⁿ.

Trick to Always Use

  • Step 1 → Separate the series into odd and even positions.
  • Step 2 → Test each half for A.P. (difference constant) or G.P. (ratio constant).
  • Step 3 → If neither fits perfectly, check for alternating add-multiply patterns.
  • Step 4 → Use sub-series logic to find the next term in each group.
  • Step 5 → Always verify by recombining to ensure consistency.

Summary

Summary

  • Mixed Series combine additive (A.P.) and multiplicative (G.P.) patterns.
  • Separating odd and even terms is often the fastest way to reveal the rule.
  • Common mixed forms include alternate add-multiply or sub-series with independent rules.
  • Always confirm the rule by calculating both differences and ratios.
  • Apply the same sub-rule forward to find missing or next terms.

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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