# Number Series Shortcuts

In questions based on number series, some numbers are arranged in a particular sequence. You have to decipher that particular sequence of numbers and on the basis of that deciphered sequence, find out the next number of the series. Although there is no limit of logics which can be used to build a series, here are some important examples given which highlight the type of series asked in the examination.

### How to solve number series problems:

Step 1 Observe if there are any familiar numbers in the given series.  Familiar numbers are primes numbers, perfect squares, cubes ... which are easy to identify.
Examples:
aPrime number series: 2, 3, 5, 7, 11, 13, 17, . . .
bSquare series : 1, 4, 9, 16, 25, 36, 49, . . .
cCube series: 1, 8, 27, 64, 125, 216, 343, 512, 729, . . .
d $${n^2} - 1$$ series:  0, 3, 8, 15, 24,35, 48,
e $${n^2} + n$$ series: 2, 6, 12, 20, . . . .
f $n \times (n+1)$ series: 1 × 2, 2 × 3, 3 × 4, 4 × 5. . .

Step 2Calculate the differences between the numbers.  Observe the pattern in the differences.  If the differences are growing rapidly it might be a square series, cube series, or multiplicative series.  If the numbers are growing slowly it is an addition or subtraction series.
If the differences are not having any pattern then
1.  It might be a double or triple series.  Here every alternate number or every 3rd number form a series
2.  It might be a sum or average series.  Here sum of two consecutive numbers gives 3rd number.  or average of first two numbers give next number.

Example 1: 12, 9, 17, 13, 22, 17, 27, 21, 32, 25, 37, . . .
Answer: The above series is a mixed series.  12, 17, 22, 27, 32, 37, . . . form a series, 9, 13, 17, 21, 25, . . . form another series.  So after 37 we get 29.

Example 2: 720, 120, 24, 6, ?
Answer: 720/6 = 120, 120/5 = 24, 24/4 = 6, 6/3 = 2

Example 3: 32, 48, 72, 108, ?
48/32 = 3/2.  So Number × 3/2= next number. 32 × 3/2 = 48, 48 x 3/2 = 72, 72 × 3/2 = 108,  108 × 3/2 = 162.

Example 4: 2, 6, 18, 54, 162, ?
The numbers are multiplied by 3 to get next number. (162 × 3 = 486).

Step 3If nothing works, then check image series, average series, etc.
Example: 34, 81, 72, 47, 74, 27, 18, ?
Answer :  Look at the middle two numbers. (47, 74).  They are images.  Similarly (72, 27), (81, 18) are images. So Image of 34 is 43.

Important:Solve the exercises given to get more understanding of various patterns asked in exams.