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Fibonacci Type Series

Introduction

In a Fibonacci Type Series, each new number is formed by adding (or occasionally combining) the previous one or two numbers. This pattern tests your ability to recognize sequences where every term depends on those before it.

The most common Fibonacci pattern is when each term equals the sum of the two preceding terms. For example: 1, 1, 2, 3, 5, 8, 13... - every term is the sum of the last two numbers.

Pattern: Fibonacci Type Series

Pattern

The key idea: each number is obtained by adding the previous one or two numbers.

In the standard Fibonacci pattern:

Tₙ = Tₙ₋₁ + Tₙ₋₂
Example: if the last two numbers are 3 and 5, the next term = 3 + 5 = 8.

In advanced variations, the next term may come from:

  • Adding the last three numbers (Tribonacci type)
  • Adding and subtracting alternately
  • Or multiplying one term before adding a constant

The goal is to find a consistent rule that links every new term to one or more of the preceding ones.

Step-by-Step Example

Question

Find the next term in the series: 2, 3, 5, 8, 13, ?

Solution

  1. Step 1: Observe recent terms

    Look at the last two numbers: 8 and 13.

  2. Step 2: Identify the pattern

    Check if each term equals the sum of the previous two:
    2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13.
    The rule follows Tₙ = Tₙ₋₁ + Tₙ₋₂.

  3. Step 3: Apply the rule

    Next term = 8 + 13 = 21.

  4. Final Answer:

    21

  5. Quick Check:

    Verify previous steps: 2+3=5, 3+5=8, 5+8=13, 8+13=21 ✅

Quick Variations

1. Standard Fibonacci: Sum of the last two numbers.

2. Tribonacci: Sum of the last three numbers (e.g., 1, 2, 3, 6, 11, 20...).

3. Mixed Pattern: Add then subtract alternately (e.g., +, -, +, -).

4. Multiplicative Mix: Multiply one term before adding a constant (e.g., ×2 + 3).

5. Offset Pattern: Add the first and third previous terms (e.g., Tₙ = Tₙ₋₁ + Tₙ₋₃).

Trick to Always Use

  • Step 1 → Check if the current number equals the sum of the two before it.
  • Step 2 → If not, test the sum of the last three numbers (Tribonacci).
  • Step 3 → Look for alternating addition/subtraction or small constants being added.
  • Step 4 → Always verify the pattern across multiple terms before deciding.

Summary

Summary

  • Fibonacci Series → Next term = Sum of the previous two.
  • Tribonacci or higher → Next term = Sum of 3 or more previous terms.
  • Some versions use alternation (add then subtract) or multiplication.
  • Always confirm the rule works consistently through all terms.

Example to remember: 2, 3, 5, 8, 13, 21 → each term = sum of the two before it.

Practice

(1/5)
1. Find the missing term: ?, 2, 3, 5, 8
easy
A. 1
B. 0
C. 4
D. 2

Solution

  1. Step 1: Look at the known terms

    Sequence (from second term): 2, 3, 5, 8.

  2. Step 2: Use reverse Fibonacci

    If Tₙ = Tₙ₋₁ + Tₙ₋₂ then earlier term = later - previous. So missing = 3 - 2 = 1.

  3. Final Answer:

    1 → Option A

  4. Quick Check:

    1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 ✅
Hint: Work backward: subtract to find the earlier term.
Common Mistakes: Trying to add forward instead of subtracting for missing earlier term.
2. Find the next term: 2, 3, 6, 10, 17, ?
easy
A. 26
B. 28
C. 30
D. 29

Solution

  1. Step 1: Test a modified Fibonacci rule

    Check if Tₙ = Tₙ₋₁ + Tₙ₋₂ + 1.

  2. Step 2: Verify initial steps

    2+3+1=6, 3+6+1=10, 6+10+1=17 - rule holds.

  3. Step 3: Compute next term

    Next = 10 + 17 + 1 = 28.

  4. Final Answer:

    28 → Option B

  5. Quick Check:

    Sequence: 2,3,6,10,17,28 → each term = prev + one-before-prev + 1 ✅
Hint: When sums are slightly larger than plain Fibonacci, check for a constant +1/+2 offset.
Common Mistakes: Assuming pure Fibonacci without the small constant offset.
3. Find the next term: 4, 6, 10, 16, 26, ?
easy
A. 40
B. 41
C. 42
D. 43

Solution

  1. Step 1: Test classic Fibonacci addition

    Check whether each term = sum of the previous two.

  2. Step 2: Verify

    4+6=10, 6+10=16, 10+16=26 - rule holds.

  3. Step 3: Compute next

    Next = 16 + 26 = 42.

  4. Final Answer:

    42 → Option C

  5. Quick Check:

    4,6,10,16,26,42 - simple Fibonacci-type addition ✅
Hint: If differences grow, try adding the last two terms.
Common Mistakes: Looking for constant differences instead of summing.
4. Find the next term: 1, 2, 3, 6, 9, 16, ?
medium
A. 25
B. 24
C. 26
D. 27

Solution

  1. Step 1: Identify alternating rule

    Here odd-index terms follow Tₙ = Tₙ₋₁ + Tₙ₋₂, while even-index terms use Tₙ = Tₙ₋₁ + Tₙ₋₂ + 1.

  2. Step 2: Verify

    1+2=3 (odd), 2+3+1=6 (even), 3+6=9 (odd), 6+9+1=16 (even) - pattern holds.

    3. Step 3: Compute next (odd index)

    Next = 9 + 16 = 25.

  3. Final Answer:

    25 → Option A

  4. Quick Check:

    1,2,3,6,9,16,25 → alternation of +0/+1 offset verified ✅
Hint: Check for alternating small offsets on even/odd positions.
Common Mistakes: Assuming a single uniform rule for all positions.
5. Find the next term: 2, 3, 7, 13, 27, 53, ?
medium
A. 100
B. 101
C. 102
D. 107

Solution

  1. Step 1: Test a multiplicative blend rule

    Try Tₙ = Tₙ₋₁ + 2×Tₙ₋₂.

  2. Step 2: Verify

    2 and 3 → 3 + 2×2 = 7, 7 + 2×3 = 13, 13 + 2×7 = 27, 27 + 2×13 = 53 - rule holds.

  3. Step 3: Compute next

    Next = 53 + 2×27 = 53 + 54 = 107.

  4. Final Answer:

    107 → Option D

  5. Quick Check:

    Sequence: 2,3,7,13,27,53,107 - each term = previous + 2×one-before-prev ✅
Hint: If sums grow faster than Fibonacci, try small multiplicative blends (e.g., +2×previous-2).
Common Mistakes: Forcing pure Fibonacci when growth suggests multiplication.

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