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Sequence and Series

To crack any question of series, try the following tricks in the same order as mentioned below –
  1. Squares/Cubes: First of all try to see whether the series has been made from the squares/cubes of natural numbers. Most of the time series is made by adding or subtracting a fix value from the squares/cubes of natural numbers.
     
    For Example

    Find the next term in the series- 2, 9, 28, 65, ? 

    In this example, each term has been constructed by adding 1 to the cubes of natural number starting with 1. ie. we can write the above series as (13 + 1), (23 + 1), (33 + 1), (43 + 1), hence the next term should be (53 + 1) or 126.
     

  2. Arithmetic/Geometric Progression: Next you, have to see if the series is an arithmetic progression or a geometric progression. In many case it will be.
     
    Arithmetic progression is made by adding or subtracting a fix value from each consecutive term.
     
    For Example

    4, 11, 18, 25 is an AP, in which, each new term has been formed by adding 7 to previous term. So, in this case next term will be 32.

    Similarly, 6, 36, 216, 1296 is an GP which has been created by multiplying 6 to each term. Next term in this case will be 7776.
     

  3. Mix of AP/GP: Many times, series are formed by combining AP and GP together. For example, let find the next term of the following series – 2, 4, 12, 48, ?
     
    In this series, each next term has been formed by multiplying a number (Variable) in the previous term and the multiplying number is increasing by a value 1 each time.
     
    So, initially 2 was multiplied by 2, then 4 has been multiplied by 3, then 12 has been multiplied by 4, hence 48 should be multiplied by 5, which gives us the answer as 240.
  4. Finding Answer by taking the difference up to two levels: Most of the series questions which do not fall under above categories, can be solved by taking the difference of each term up to 2 levels.
     
    Example
    3, 6, 11, 18, 27, ?
    Solution
    When we subtract the numbers of the series for the first time, then we get 3 (6 - 3), 5 (11 - 6), 7 (18 - 11), 9 (27 - 18), So, the next term will be 38 (38 - 27 = 11)..
     
  5. Miscellaneous Series:
    1. Find the next term in the series – 2, 3, 5, 7, 11, 13, ?
       
      This is a series formed by all prime numbers. Hence the answer is 17.
    2. Find the next term in the series – 77, 49, 36, 18, ?
       
      In this series, multiplication of the digits of any number forms next number in the series. Hence next number should be 8
    3. Find the next term in the series – 2, 5, 7, 12, 19, 31, ?
       
      In this series, each term has been formed by adding the previous two terms. Ie. 7 = 2 + 5, 12 = 5 + 7, and so on. Hence in this case the last term will be 50
    4. Find the next term of the series – 3, 4, 12, 48, ?
       
      In this series, each term has been formed by multiplying the previous two terms. ie. 12 = 3 × 4, 48 = 4 × 12 and so on. Hence in this case the last term will be 576




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