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Alternating Pattern Series

Introduction

An Alternating Pattern Series contains two or more interleaved sequences, each following its own rule. These appear frequently in reasoning tests where odd and even positions (or grouped positions) form independent sub-series and must be analysed separately.

Correctly splitting the series and solving each sub-series is the key to finding the missing or next term accurately.

Pattern: Alternating Pattern Series

Pattern

The key idea: Different positions (odd/even or grouped indices) follow distinct progressions - treat each as its own sequence and then recombine.

Formula:
If the series alternates between two arithmetic progressions:
aodd,n = a1 + (n - 1)·d1
aeven,n = a2 + (n - 1)·d2
where d1 and d2 are the common differences for odd and even positions respectively.

Tip: Separate the sequence by index (1st, 3rd, 5th → sub-series A; 2nd, 4th, 6th → sub-series B), then solve each sub-series independently.

Step-by-Step Example

Question

Find the next term in the series: 2, 4, 3, 6, 4, 8, 5, ?

Solution

  1. Step 1: Split into sub-series

    Odd-position terms (1st, 3rd, 5th, 7th): 2, 3, 4, 5.
    Even-position terms (2nd, 4th, 6th, 8th): 4, 6, 8, ?.

  2. Step 2: Identify patterns for each sub-series

    Odd sub-series: 2 → 3 → 4 → 5 → difference = +1 (AP with d₁ = 1).
    Even sub-series: 4 → 6 → 8 → ? → difference = +2 (AP with d₂ = 2).

  3. Step 3: Apply the rules to find the next term

    The next term in the even sub-series = 8 + 2 = 10.
    Since the unknown is at the 8th position (even), use the even sub-series result.

  4. Final Answer:

    10

  5. Quick Check:

    Odd terms: 2, 3, 4, 5 (↑ +1). Even terms: 4, 6, 8, 10 (↑ +2). Combined: 2,4,3,6,4,8,5,10 ✅

Quick Variations

1. Two APs: odd and even terms both arithmetic but with different d (example above).

2. AP & GP mix: one sub-series arithmetic, the other geometric (e.g., odd: +2, even: ×2).

3. Three-way interleave: positions 1,4,7... follow rule A; 2,5,8... follow rule B; 3,6,9... follow rule C.

4. Alternating transforms: one sub-series may be n², the other n³ or n² ± 1.

Trick to Always Use

  • Step 1: Separate the series by position (odd/even or groups of positions).
  • Step 2: Identify the rule in each sub-series (AP, GP, n², etc.).
  • Step 3: Extend only the sub-series matching the position of the missing term.

Summary

Summary

  • Alternating series hide multiple independent rules - always separate by index first.
  • Solve each sub-series independently using AP/GP or other known patterns.
  • Apply the correct rule based on whether the missing term is in an odd or even position.
  • Check both sub-series after solving to verify pattern consistency.

Example to remember:
3, 6, 5, 10, 7, 14, ? → Odd terms +2; Even terms +4 → Next = 9

Practice

(1/5)
1. Find the next term in the series: 1, 3, 2, 6, 3, 9, 4, ?
easy
A. 12
B. 11
C. 13
D. 14

Solution

  1. Step 1: Split into sub-series

    Odd-position terms: 1, 2, 3, 4. Even-position terms: 3, 6, 9, ?.

  2. Step 2: Identify patterns

    Odd sub-series increases by +1. Even sub-series increases by +3 (3 → 6 → 9).

  3. Step 3: Apply the rule

    Next even-term = 9 + 3 = 12. The missing term is at the 8th (even) position, so answer = 12.

  4. Final Answer:

    12 → Option A

  5. Quick Check:

    Odd: 1,2,3,4 (+1); Even: 3,6,9,12 (+3) → Combined sequence holds ✅
Hint: Split into odd/even positions and check small constant steps (+1, +2, +3...).
Common Mistakes: Trying to apply a single rule to the whole sequence instead of splitting.
2. Find the next term in the series: 4, 7, 5, 8, 6, 9, 7, ?
easy
A. 10
B. 9
C. 11
D. 12

Solution

  1. Step 1: Split into sub-series

    Odd-position terms: 4, 5, 6, 7. Even-position terms: 7, 8, 9, ?.

  2. Step 2: Identify patterns

    Odd sub-series: +1 each time. Even sub-series: +1 each time.

  3. Step 3: Apply the rule

    Next even-term = 9 + 1 = 10. The missing term is 8th (even), so answer = 10.

  4. Final Answer:

    10 → Option A

  5. Quick Check:

    Odd: 4,5,6,7; Even: 7,8,9,10 → Combined holds ✅
Hint: When both sub-series use the same small increment, extension is straightforward by parity.
Common Mistakes: Overcomplicating with alternate multipliers when simple +1 works.
3. Find the next term in the series: 10, 2, 11, 4, 12, 6, 13, ?
easy
A. 7
B. 8
C. 9
D. 14

Solution

  1. Step 1: Split into sub-series

    Odd-position terms: 10, 11, 12, 13. Even-position terms: 2, 4, 6, ?.

  2. Step 2: Identify patterns

    Odd sub-series: +1 each time. Even sub-series: +2 each time (2 → 4 → 6).

  3. Step 3: Apply the rule

    Next even-term = 6 + 2 = 8. The missing term is 8th (even), so answer = 8.

  4. Final Answer:

    8 → Option B

  5. Quick Check:

    Odd: 10,11,12,13; Even: 2,4,6,8 → Combined holds ✅
Hint: Label odd/even subsequences and look for simple +1/+2 patterns.
Common Mistakes: Mixing values across sub-series when predicting the next term.
4. Find the next term in the series: 3, 6, 5, 10, 7, 14, 9, ?
medium
A. 15
B. 16
C. 18
D. 20

Solution

  1. Step 1: Split into sub-series

    Odd-position terms: 3, 5, 7, 9. Even-position terms: 6, 10, 14, ?.

  2. Step 2: Identify patterns

    Odd sub-series: increases by +2. Even sub-series: increases by +4 (6 → 10 → 14).

  3. Step 3: Apply the rule

    Next even-term = 14 + 4 = 18. The missing term is 8th (even), so answer = 18.

  4. Final Answer:

    18 → Option C

  5. Quick Check:

    Odd: 3,5,7,9 (+2); Even: 6,10,14,18 (+4) → Combined holds ✅
Hint: When gaps differ (e.g., +2 vs +4), label sub-series and extend the correct one by index.
Common Mistakes: Assuming a single uniform gap for the entire series.
5. Find the next term in the series: 7, 14, 8, 16, 9, 18, 10, ?
medium
A. 19
B. 20
C. 21
D. 22

Solution

  1. Step 1: Split into sub-series

    Odd-position terms: 7, 8, 9, 10. Even-position terms: 14, 16, 18, ?.

  2. Step 2: Identify patterns

    Odd sub-series: +1 each time. Even sub-series: +2 each time (14 → 16 → 18).

  3. Step 3: Apply the rule

    Next even-term = 18 + 2 = 20. The missing term is 8th (even), so answer = 20.

  4. Final Answer:

    20 → Option B

  5. Quick Check:

    Odd: 7,8,9,10 (+1); Even: 14,16,18,20 (+2) → Combined holds ✅
Hint: Always check which sub-series the missing index belongs to before calculating.
Common Mistakes: Swapping odd and even subsequences or mixing their rules.

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