Solving Series Problems

Three types series questions are asked in exams:

1. Alphabetical Type
2. Number Type
3. Mixed Type


Part 1: Solving Alphabetical Series Problems

In such questions of alphabetic series, if the question consists of a single alphabet series, then you have to solve the logic implied in the sequence. Then fill in the missing character with a correct choice.

Note: Remember all the alphabets and their place number ,You can even note down during your exam.

A 1 26
B 2 25
C 3 24
D 4 23
E 5 22
F 6 21
G 7 20
H 8 19
I 9 18
J 10 17
K 11 16
L 12 15
M 13 14
N 14 13
O 15 12
P 16 11
Q 17 10
R 18 9
S 19 8
T 20 7
U 21 6
V 22 5
W 23 4
X 24 3
Y 25 2
Z 26 1

There are 26 letters in English alphabets.
The first half of alphabet series is from A to M and the half is from N to Z.

A-M – 1-13 (First Alphabetical Half)

N-Z –14-26 (Second Alphabetical Half)

If you remember these intervals,It would be helpful in solving alphabet series problems.


There are 3 main types of Alphabetical series

Type 1: Alphabet Series

Increasing by a definite number
For Example: IJKL? ( each letter increases by 1)

AGMSY? ( each letter increases by 6 place to its right position)

Decreasing by a definite number
For Example: ZXVTRP ? ( each letter decreases by 2 places to its left )

Increasing successively
For Example: DEGJNS? ( +1,+2,+3,+4,+5)

Decreasing successively
For Example: ZYWTP ( -1,-2,-3,-4 ..)

ZTOKHFE ( -6,-5,-4,-3,-2,-1)

Decreasing and Increasing by a constant value.
For Example: DFCEBDACZ (+2,-3,+2,-3,…)


Type 2: Alphanumeric Series

Example: Z1A, X2D,V6G,T21J,R88M, P445P,?

First letter: ZXVTRP (-2,-2,-2,…..)

Second letter: ADGJMP ( +3, +3,+3,…)

Series of numerals: 1,2,6,21,88,445 ( x1+1, x2+2, x3+3…)

So next term is N2676S.

Example: 2Z5,7Y7,14X9,23W11,34V13,?

First numeral- 2,7,14,23,34 (+5,+7,+9,+11..)

Second letter- ZYXWV ( decreases by 1 each time)

Third numeral- 5,7,9,11,13 ( increases by 2 each time)

Example: W-144 , U-121, S-100, Q-81,?

First letter- decreases by 2 each time

Second numeral- square of 12,11,10,9,8..


Type 3: Continuous Patterns Series

Example: ab_ _ baa_ _ ab_

Options: i) aaaaa ii) aabaa iii) caabab iv) baabb

Solution: our answer is ii) . Here series aba is repeated

Example: ab_aa_bbb_aaa_bbba

Options: i) abba ii) baab iii) aabb iv) abab

Solution: our answer is ii) . The series is abb/aaabbb/aaaabbbb/a. Thus the letter are repeated twice , then thrice , then four times and so on .

Example: bc_ca_aba_c_ca

Options i)abcbb ii)bbbcc iii)bacba iv)abbcc

Solution: our answer is i) . The series is abc/bca/cab/abc/bca. Thus the letter change in cyclic order .

Example: c_bd_cbcda_a_db_a

Options: i) adabcd ii) bdbcba iii) cdbbca iv)daabbc

Solutions: our answer is i). The series is acdb/dacb/cdab/acdb/da. Each group of four letters contains the letters of the previous group in the order – third , first , second and fourth.

Example: a_bb_baa_bbb_aa_

Options: i) aabba ii) bbaab iii)abaaa iv)baabb

Solutions: our answer is iii). The series is aabbbb/aaabbb/aaaa. At each step , the number of a’s increases by one while the number of b’s decrease by one.


Part 2: Solving Number Series Problems

Number series questions are based on numerical sequences that follow a logical rule/ pattern based on elementary arithmetic concepts.
A particular series is given from which the pattern must be analyzed. You are then asked to predict the next number in the sequence following the same rule. Generally, there are three types of questions asked from the number series:

A numerical series is given in which a number is wrongly placed. You are asked to identify that particular wrong number.
A numerical series is given in which a specific number is missing. You are required to find out that missing number.
A complete numerical series is followed by an incomplete numerical series. You need to solve that incomplete numerical series in the same pattern in which the complete numerical series is given.

Different types of Number Series:
The most common patterns followed by number series are:

Series consisting of Perfect Squares:
A series based on Perfect squares is most of the times based on the perfect squares of the numbers in a specific order & generally one of the numbers is missing in this type of series.

Example: 324, 361, 400, 441,?
Sol: 324 = 182 , 361 = 192, 400 = 202, 441 = 212, 484 = 222

Perfect Cube Series:
It is based on the cubes of numbers in a particular order and one of the numbers is missing in the series.

Example:512, 729, 1000,?
Sol:83, 93, 103, 113

Geometric Series:
It is based on either descending or ascending order of numbers and each successive number is obtained by dividing or multiplying the previous number by a specific number.

Example:4, 36, 324, 2916?
Sol:4 x 9 = 36, 36 x 9 = 324, 324 x 9 = 2916, 2916 x 9 = 26244.

Arithmetic Series:
It consists of a series in which the next term is obtained by adding/subtracting a constant number to its previous term. Example: 4, 9, 14, 19, 24, 29, 34 in which the number to be added to get the new number is 5. Now, we get an arithmetic sequence 2,3,4,5.

Two-stage Type Series:
In a two step Arithmetic series, the differences of consecutive numbers themselves form an arithmetic series.
Example: 1, 3, 6, 10, 15.....
Sol:3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4, 15 - 10 = 5....
Now, we get an arithmetic sequence 2, 3, 4, 5

Mixed Series:
This particular type of series may have more than one pattern arranged in a single series or it may have been created according to any of the unorthodox rules.

Example:10, 22, 46, 94, 190,?
Sol:
10 x 2 = 20 +2 = 22,
22 x 2 = 44 + 2 = 46,
46 x 2 = 92 + 2 = 94,
94 x 2 = 188 + 2 = 190,
190 x 2 = 380 + 2 = 382.
So the missing number is 382.

Arithmetico –Geometric Series :
As the name suggests, this type of series is formed by a peculiar combination of Arithmetic and Geometric series. An important property of Arithmetico- Geometric series is that the differences of consecutive terms are in Geometric Sequence.

Example:1, 4, 8, 11, 22, 25, ?
Sol :Series Type +3 , X2 ( i.e Arithmetic and Geometric Mixing)
1 + 3 = 4, 4 X 2 = 8, 8 + 3 = 11, 11 X 2 = 22, 22 + 3 = 25, 25 X 2 = 50

Geometrico - Arithmetic Series is the reverse of Arithmetico - Geometric Series. The differences of suggestive terms are in Arithmetic Series.

Example: 1, 2, 6, 36, 44, 440, ?
Sol :Series Type - X 2, + 4, X 6, +8 , X 10
1 X 2 = 2, 2 + 4 = 6, 6 X 6 = 36, 36+ 8 = 44, 44 X 10 = 440, 440 + 12 = 452

Twin/Alternate Series :
As the name of the series specifies, this type of series may consist of two series combined into a single series. The alternating terms of this series may form an independent series in itself.

Example: 3, 4, 8, 10, 13, 16 ? ?
Sol: As we can see, there are two series formed
Series 1 : 3, 8, 13 with a common difference of 5
Series 2 : 4, 10, 16 with a common difference of 6
So, next two terms of the series should be 18 & 22 respectively.


How To Solve Number Series Problems

Step 1: Familier numbers are
primes numbers, perfect squares, cubes … which are easy to identify.

Step 2: Calculate 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 substration 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

Step 3: Sometimes the number will be multiplied and will be added another number So we need to check those patterns

Types

I. Prime number Series :
Example
(1) : 2,3,5,7,11,13, ………..
Answer
: The given series is prime number series . The next prime number is 17.

Example
(2) :2,5,11,17,23,………..41.
Answer:
The prime numbers are written alternately.

2. Difference Series :
Example (1): 2,5,8,11,14,17,………..,23.
Answer:
The difference between the numbers is 3. (17+3 = 20)

Example (2): 45,38,31,24,17,………..,3.
Answer: The difference between the numbers is 7. (17-7=10).

3. Multiplication Series:
Example (1) : 2,6,18,54,162,………,1458.
Answer: The numbers are multiplied by 3 to get next number.
(162×3 = 486).

Example:
(2) : 3,12,48,192,…………,3072.
Answer : The numbers are multiplied by 4 to get the
next number. (192×4 =768).

4. Division Series:
Example (1): 720, 120, 24, ………,2,1
Answer: 720/6=120, 120/5=24, 24/4=6, 6/3=2, 2/2=1.

Example (2) : 32, 48, 72, 108, ………., 243.
Answer: 2. Number x 3/2= next number. 32×3/2=48, 48×3/2=72,
72×3/2=108, 108×3/2=162.

5. n2 Series:

Example(1) : 1, 4, 9, 16, 25, ……., 49
Answer: The series is 12, 22, 32, 42, 52, …. The
next number is 62=36;

Example: (2) : 0, 4, 16, 36, 64, …….. 144.
Answer :The series is 02, 22, 42, 62, etc. The next number
is 102=100.

6. n2?1 Series

Example: 0, 3, 8, 15, 24,35, 48, ……….,
Answer
The series is 12-1, 22-1, 32-1 etc. The next number is 82-1=63.

Another logic
Difference between numbers is 3, 5, 7, 9, 11, 13 etc. The next number is
(48+15=63).

7.n2+1 Series

Example: 2, 5, 10, 17, 26, 37, ………., 65.
Answer : The series is 12+1, 22+1, 32+1 etc. The next number
is 72+1=50.

8. n2+n Series
(or) n2?n Series

Example : 2, 6, 12, 20, …………, 42.
Answer
The series is 12+1, 22+2, 32+3, 42+4 etc. The next number = 52+5=30.

Another Logic
The series is 1×2, 2×3, 3×4, 4×5, The next number is 5×6=30.

Another Logic
The series is 22-2, 32-3, 42-4, 52-5, The next number is 62-6=30.

9. n3 Series

Example:
1, 8, 27, 64, 125, 216, ……… .
Answer
The series is 13, 23, 33, etc. The missing number is 73=343.

10. n3+n Series

Example: 2, 9, 28, 65, 126, 217, 344, ………..

Answer
The series is 13+1, 23+1, 33+1, etc. The missing number is 83+1=513.

11. n3?1 Series

Example : 0, 7, 26, 63, 124, …………, 342.
Answer: The series is 13-1, 23-1, 33-1 etc The missing
number is 63-1=215.

12. n3+n Series

Example:
2, 10, 30, 68, 130, ………….., 350.
Answer
The series is 13+1, 23+2, 33+3 etc The missing number is 63+6=222.

13. n3?n Series

Example:
0, 6, 24, 60, 120, 210, …………..,
Answer
The series is 13-1, 23-2, 33-3, etc. The missing number is 73-7=336.
Another Logic
The series is 0x1x2, 1x2x3, 2x3x4, etc. The missing number is 6x7x8=336.

14. n3+n2 Series

Example:
2, 12, 36, 80, 150, …………,
Answer:
The series is 13+12,23+22,33+32etc. The missing number is 63+62=252

15. n3?n2 Series:
Example:
0,4,18,48,100,……………..,
Answer
The series is 13-12,23-22,33-32 etc. The missing number is 63-62=180

16. xy, x+y Series:
Example:
48,12,76,13,54,9,32,……………,
Answer
2. 4+8=12, 7+6=13, 5+4=9 .: 3+2=5.


Part 3: Solving Mixed Series Problems

Mixed series starting position is always from left hand side and ending position is right side.

Again, remember of position of all letters in the alphabet, from A to Z.

A B C D E F G H I J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13

N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26


Points to remember:

Right object – Right object = Right object.
Example : Twelve object – sixth object = Sixth object from right is F.

Right object + Left object = Right object.
Example : Seventh from right object + third from left object = Tenth object from right is Q.

Left object – Left object = Left object.
Example : Fifteenth object from left side – sixth object from left side = ninth object from left is I.

Left object + Right object = Left object.
Example : Fourteenth from left object + sixth from right object = Twentieth object from left is T.

Example: Look at the series of below arrangement that which should be 11th to the right of 3rd from right end.
P * 8 S T S @ T W K L 9 # 2 9 P D B X L J N 8 F D #
Answer
11th – 3rd = 8th
So, the letter is T.