Shortcuts in Reasoning for Competitive Exams – Series
A series is a sequence of numbers/alphabetical letters or both which follow a particular rule. Each element of series is called ‘term’. We have to analyse the pattern and find the missing term or next term to continue the pattern.
Types of series are explained in the following chart:
Number series is a form of numbers in a certain sequence, where some numbers are mistakenly put into the series of numbers and some number is missing in that series, we need to observe first and then find the accurate number to that series of numbers.
• Even and odd numbers.
• Prime and composite numbers.
• Square and square roots of a numbers.
• Cube and cube roots of a numbers.
Types of Number Series
1. Perfect Square Series
This type of series are based on square of a number which is in same order and one square number is missing in that given series.
EXAMPLE 841, ?, 2401, 3481, 4761
2. Perfect Cube series
Perfect Cube series is a arrangement of numbers is a certain order, where some number which is in same order and one cube is missing in that given series.
EXAMPLE 4096,4913,5832, ?, 8000 Sol. 163, 173,183, 193,203
3. Mixed number series
Mixed number series is a arrangement of numbers in a certain order. This type of series are more than are different order which arranged in alternatively in single series or created according to any non conventional rule.
EXAMPLE 6, ?, 33, 69, 141, 285
Sol. x2 + 3, x2 + 3, x2 + 3, x2 + 3, x 2 + 3, x 2 + 3
4. Geometric Series Geometric Number series is a
arrangement of numbers in a certain order, where some numbers are this type of series are based on ascending or descending order of numbers and each continues number is obtain by multiplication or division of the previous number with a static number.
In geometric series number is a combination of number arranged.
EXAMPLE 21, 84, 336, ?, 5376 Sol. 21 x 4 = 84 84 x 4 = 336 336 x 4 = 1344 1344 x 4 = 5376
5. Prime series
When numbers are a series of prime numbers.
EXAMPLE 2,3, 5, 7, 11, 13, _ , 19
Solution. Here, the terms of the series are the prime numbers in order. The prime number, after 13 is 17. So, the answer to this question is 17.
6. Alternate Primes
It can be explained by below example.
EXAMPLE 2, 11, 17, 13, _, 41
Solution. Here, the series is framed by taking the alternative prime numbers. After 23, the prime numbers are 29 and 31. So, the answer is 31.
7. The difference of any term from its succeeding term is constant (either increasing series or decreasing series):
EXAMPLE 4,7,10,13,16,19, _, 25
Solution. Here, the differnce of any term from its succeding term is 3. 7-4 = 3 10 – 7 = 3
So, the answer is 19 + 3 = 22
8. The difference between two consecutive terms will be either increasing or decreasing by a constant number:
EXAMPLE 2, 10, 26, 50, 82,
Solution. Here, the difference between two consecutive terms are
10-2 = 8 26-10= 16 50 – 26 = 24
82 – 50 = 32
Here, the difference is increased by 8 (or you can say the multiples of 8). So the next difference will be 40 (32 + 8). So, the answer is 82 + 40= 122
9. The difference between two numbers can be multiplied by a constant number:
EXAMPLE 15, 16, 19, 28, 55, _
Solution. Here, the differences between two numbers are 16- 15 = 1 19-16 = 3 28-19 = 9 55 – 28 = 27
Here, the difference is multiplied by 3. So, the next difference will be 81. So, the answer is 55 + 81 = 136
10. The difference can be multiples by number which will be increasing by a constant number:
EXAMPLE 2, 3, 5, 11, 35,_
Solution. The difference between two number are 3-2=1 5-3=2 11-5 = 6 35 – 11 = 24
11. Every third number can be the sum of the preceding two numbers :
EXAMPLE 3, 5, 8, 13, 21, _
Solution. Here, starting from third number 3 + 5 = 8 5 + 8 = 13 8 + 13 = 21
So, the answer is 13 + 21 = 34
12. Every third number can be the product of the preceeding two numbers :
EXAMPLE 1, 2, 2, 4, 8, 32. _
Solution. Here, starting from the third number
1×2 = 2 2×2 = 4 2×4 = 8
4 x 8 = 32
So, the answer is 8 x 32 = 256
13. Every succeeding term is got by multiplying the previous term by a constant number or numbers which follow a special pattern.
EXAMPLE 5, 15, 45, 135, _
5 x 3 = 15 15 x 3 = 45 45 x 3 = 135
So, the answer is 135 x 3 = 405
14. In certain series the terms are formed by various rule (miscellaneous rules). By keen observation you have to find out the rule and the appropriate answer.
EXAMPLE 4, 11, 31, 90, _
Solution. Terms are,
4×3-1 = 11 11 x 3-2 = 31 31×3 – 3 = 90
So, the answer will be 90 x 3 – 4 = 266
- First check the direct formulas.
- If all the numbers are even, odd or I prime.
- If all the number are perfect squares ‘ or cubes.
- If all the numbers have a particular 1 divisibility.
- If all the numbers are succeeding . by some additions or subtraction 1 or multiplications or divisions by a I particular number or addition of I their cubes and squares.
- When the difference between the consecutive numbers is same/ constant or the number series is in
a, a + d, a + 2d,…, a + ( n – 1) d.Where ‘a’ is first term, d is the common difference.
- When any number series is in the forma,a+(a + l),a+(a+l) + (a+2), … , nth term of the series beShortcut Approach
- If numbers are in ascending order in the number series.
Numbers may be added or multiplied by certain numbers from the first number.
If numbers are in descending order in the number series Numbers may be subtracted or divided by certain numbers from the first number.
If numbers are in mixing order (increasing and decreasing) in the number series.
Numbers may be in addition,subtraction, multiplication and division in the alternate numbers.
Step 1 : Check whether it is ascending, descending or mixed order.
Step 2 : It is in mixing order. So it may be in addition, subtraction, division and multiplication, squares and cubes.
Step 3 : In above series it is mixing of .square, addition and subtraction.
(14)2= 196 + 4 = 200
(13)2 = 169. By adding 4 it gives 173.
Try subtraction. ‘
169-4 = 165
Here we found it is in order of squaring a number, adding by 4 and subtracting by 4.
Hence, the answer for above series is 77.
A series that is made by only alphabetic letters.
EXAMPLE G, H, J, M, ?
Remember all the alphabets and their place number.
Intervals like :
ALPHA NUMERIC SERIES
These kind of problems used both mathematical operation and position of letters in the alphabet in forward, backward order.
EXAMPLE K 1, M 3, P 5, T 7, ?
Solution. The given sequence consists of two series
Z, X, V, T, _
L, J, H, F,_ . Both consisting
of alternate letters in the reverse order.
Next term of (i) series = R, and Next term of (ii) series = D
CONTINUOUS PATTERN SERIES
It is a series of small/capital letters that follow a certain pattern like repetition of letters.
- Firstly, count the number of blanks and given letters.
- Note down the pattern common to all groups separately.
- Divide the whole sum of blanks and letters by a multiple.