Number Series: Mastery

1. Difference Based Series

First, find the difference between the given numbers.

• Simple Difference: Like \(3, 7, 11, 15\) (Difference \(+4\)).
• Double/Triple Difference: If the first difference is not constant, find the difference of the difference.
  Example from PDF: \(2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182\)
  First Difference: \(4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26\)
  Second Difference: \(2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2\) (Constant \(+2\)).
• Alternate/Zig-Zag Pattern: Plus-minus pattern.
  Example: \(211, 90, 171, 122, 147, 138, 139\)
  Pattern: \(-121, +81, -49, +25, -9, +1\) (Subtracting and adding consecutive odd squares).
• Prime Number Difference: Differences are always prime numbers.
  Example: \(+41, +43, +47, +53\).
• Factorial Difference: Differences are in the form of \(2!, 3!, 4!, 5!\)
  Example: \(46, 48, 54, 78, 198, 918\)

2. Multiplication & Division Based

When numbers increase rapidly, look for multiplication patterns.

• Simple Multiplier: \(\times 2, \div 3, \times 4, \div 5\)
• Multiply + Add/Sub: \(\times 1+2, \times 2+3, \times 3+4\)
• Multiply + Square/Cube: \(\times 2+2^2, \times 3+3^2, \times 4+4^2\)
• Decimal Multiplier: \(\times 0.5\) (Value decreases), \(\times 1.5, \times 2.5\)

3. Square, Cube & Special Series

• Square/Cube Patterns: \(1^3-1, 2^3-2, 3^3-3\) or \(1^3-1^2, 2^3-2^2\)
• Fibonacci Series: Sum of previous two numbers is the next number. (\(7, 9, 16, 25, 41, 66\))
• Twin Series: Two different patterns run in the same series. (\(5, 19, 45, 95, 193, 387\))
• Decimal Series: Remove the decimal point and look at the pattern. (e.g., \(12.9, 47.2, 68.8, 81.3, 87.7, 90.4\))

4. Mains Level Concepts

• Wrong Term to New Series: Find the wrong number in the first series (let's say 'P'), then start a new series from 'P' and find its 5th or 6th term.
• Equation Formation: After finding the wrong number 'n', a quadratic equation like \(x^2 + 14x + n = 0\) is formed, and you are asked to find the difference between its roots.
• Variables (P, Q, R): Multiple series are given with missing terms like P+Q, Q+R. You need to find their values and establish the relation (e.g., x > y).