Solving Number Series Questions Can Be So Simple…Learn How?

In the last article, I had discussed various types of series. Today I will discuss how to approach a series problem once you have identified what kind it is. Let me be honest. There is no algorithm for solving number series questions appearing in quantitative section of any competitive exam because each question is different and there are innumerable ways in which a series can be generated. Cracking a series question involves a lot of practice and the intuition that comes with practice.

Series Questions – Samples

  • 2, 4, 8, 16, 32, 64
  • 729, 512, 343, 216, 125, 64
  • 1, 1, 2, 4, 6, 10, 16, 26, 42, 68… (Fibonacci Series)
  • 4, 9, 6, 18, 9, 27, 13, 36

I will today broadly discuss steps how to build this intuition so that you can solve series questions easily and quickly.

Step 1: Screening

First check the series by giving a cursory look at it. Many a times a careful first look may be enough to tell the next term. Have a look at these examples:

  1. 3, -9, 27, -81, 243, ?
  2. 1, 3, 7, 13, 21, 31, ?
  3. 1, 3, 6, 10, 15, 21, ?
  4. 2, 4, 8, 16, 32, 64, ?

Solution:

  1. Each successive term is multiplied by -3. Next term will be 243 X -3= -729.
  2. The series is +2, +4, +6, +8, +10, +12. So next term will be 31 + 12 = 43.
  3. The series is +2, +3, +4, +5,+6, +7. Next term will be 21 + 7 = 28.
  4. Each term is multiplied by 2. Next term will be 64 X2 = 128.

Step 2: check the pattern: increasing/ decreasing/ alternating

In case you fail to decipher the rule of the series by just preliminary series, try to understand the trend of the series. Look at the pattern. Is it increasing or decreasing? Is it following an alternating pattern?

  1. 1, 4, 9, 16, 25, 36, 49.
  2. 4, -8, 16, -32, 64, -128.
  3.  729, 512, 343, 216, 125, 64.
  4. 5, 10, 13, 26, 29, 58.

Clearly, i and iv are following an increasing pattern. ii is following an alternating pattern while iii is a decreasing pattern series .

Step 3: if the series is increasing or decreasing, find the rate of increase or decrease

Start with the first term and move onto the next. Gauge whether the series proceeds arithmetically or geometrically or alternately. You have to feel whether this rise or fall is slow or fast. In an arithmetic progression, the increase or decrease of terms is by virtue of addition or subtraction. So the rise or fall will appear slow. In contrast, in a geometric progression the increase or decrease of terms is by virtue of multiplication or division. So naturally the rise or fall will be very fast. If the geometric progression involves squaring or cubing, this rise or fall will be even sharper. I hope you understand how you need to feel the rate of increase or decrease.

In the above examples now it is clear that 1, 4, 9, 16, 25, 36, 49,… is an arithmetic type of series. The trend being +3, +5, +7, +9 and so on. Clearly, 729, 512, 343, 216, 125, 64,… is a geometric type of series involving cubes of 9, 8, 7, 6, 5, 4 and so on.

Now look at 1, 5, 14, 30, 55, 91. It can easily be figured out that the rise is rather sharp. So it is a geometric progression. On trial you will see that just successive multiplication is not involved here. Checking for addition of squared numbers or cubed numbers we see that this is actually 12, 12+22, 12+22+32 and so on.

Basically, a geometric increase or decrease can take place in two ways –

  • Multiplication or division by terms
  • Addition or subtraction of squared or cubed terms

Obvious question is how does one differentiate between the two?

Look at the trend of the increase. If the increase is because of addition of squared or cubed numbers, the increase will be very sharp initially but not very sharp in the later terms say 5th or 6th term onwards.

Watch the earlier example:  1, 5, 14, 30, 55, 91.

1, 1X5=5, 5X2.8=14, 14X2.1428=30, 30X1.8333=55, 55X1.6545= 91.

Notice how the multiplication factor is gradually decreasing from 5 in the initial step to 1.6545 in the last step. Hence, where the rise is very sharp initially but gradually slows down, it should strike in your mind that you have to look for a pattern involving addition of squared or cubed numbers. You needn’t do the above calculations of multiplication factor, just try to build the intuition of understanding the rate of increase or decrease.

Consider 4, 5, 12, 39, 121, 610. This series also rises very sharply. Hence this must be a geometric progression. However the rate of increase does not slow down in later terms. In fact it picks up as the series progresses. Therefore we can conclude that the series must be of the first kind i.e. formed by multiplication. A little more exercise will tell us that the series is : x1+1, x2+2, x3+3, x4+4, x5+5 and so on.

Now let me show you an example of alternating increase. Two possibilities will exist here:

  • Two different series may be intermixed
  • Two different kind of operations may be being performed on successive terms

Consider the series 1, 4, 5, 9, 14, 20, 30, 43. This series increases gradually but the increase is rather haphazard. Actually it is a mix of two series!

1, 5, 14, 30, 55, 91 which is  a series: 1, 1+22, 1+22+32, 1+22+32+42 ; and 4, 9, 20, 43, 90 which is a series: x2+1, x2+2, x2+3, x2+4.

Again, look at 3, 13, 18, 76, 81, 409, 414. Here two different operations are being performed alternately: the first operation is that of multiplication by 3, 4 and 5 successively and adding a constant number 4 and the second operation is adding a constant number 5. Hence the series is x3+4, +5, x4+4, +5, x5+4, +5.

Step 4: Check if the series is neither increasing nor decreasing but alternating

For an alternating series also you should check whether it is a mix of two series or two different operations are being performed alternately.

For example, 4, 9, 6, 18, 9, 27, 13, 36. It is a mix of two series 4, 6, 9, 13 and 9, 18, 27, 36.

In 200, 600, 1200, 1600, 3200, 3600, 7200 two operations are going on; one addition of 400 and second multiplication by 2.

With practice these steps can help you build the intuition required to solve problems related to series. Did you ever face any difficulty in solving number series questions?

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Do you get terrified by Number Series Questions?

‘Number Series’ or ‘number sequence’ is an integral part of Quantitative Aptitude Section in various competitive examinations like IBPS Bank exams, SSC, RAIL and so on. I’ve come across many students who get petrified by looking at questions on number series. Lack of confidence subdues their logical faculty to come up with the missing number or next number in a series of numbers that are seemingly complicated. They are unable to decipher the predefined rule behind the sequence of numbers given. Once the rule is determined it’s simple to find out the next term(s) or missing term(s) in a series. It becomes simpler to solve number series questions following a step by step approach.

Different Type of Number Series

A series can be created in numerous ways. An understanding of these various ways can help us in recognizing the pattern followed in the number series. So here we go with some standard series types-

Arithmetic Series – Difference between successive terms is fixed. Subsequent terms are obtained by either adding or subtracting a fixed number. For example,

2, 5, 8, 11, 14, 17,…..                                Common Difference = 3

32, 25, 18, 11, 4,…….      Common Difference = -7

Geometric Series – Each term of the series is obtained by multiplying (or dividing) the previous number by a fixed number. Hence the ratio between any 2 consecutive terms is same. For example,

3, 6, 12, 24, 48, 96…….       Common Ratio = 2

2048, 512, 128, 32…….       Common Ratio = 1/4

Arithmetico-Geometric Series –  Each term is first added (or subtracted) by a fixed number and then multiplied (or divided) by another number to obtain the subsequent term. For example,

4, 18, 60, 186….. => 4, (4+2)x3, (18+2)x3, (60+2)x3

Geometrico-Arithmetic Series – Each term is first multiplied (or divided) by a fixed number and then added (or subtracted) by another number to obtain the subsequent term.For example,

3, 10, 24, 52…… => 3, (3×2)+4, (10×2)+4, (24×2)+4,…..

Series of Squares, Cubes, etc. – Each term is square or cube or a higher power of the previous term. For example,

3, 9, 81, 6561….                                         Each term is obtained by squaring the previous number

2, 8, 512, ………                                        Each term is obtained by cubing the previous number

Some non-standard ways in which series can be created –

Series with subsequent Differences being in Arithmetic Progression (AP)

3, 7, 13, 21, 31, 43……         The differences in subsequent terms are 4, 6, 8, 10, 12…. which are in AP

Series with Differences in Differences being in AP

336, 210, 120, 60, 24, 6, 0,….         The difference being 126, 90, 60, 36, 18, 6

The differences between differences being 36, 30, 24, 18, 12,…. and so on, which are in AP

Inter-Mingled Series – In this case any two of the above series are mixed in one. For example,

1, 3, 5, 1, 9, -3, 13, -11, 17,….

Odd terms (1, 5, 9, 13, 17,….) of the series are in AP, whereas even terms (3, 1, -3, -11,…) are in geometrico-arithmetic series in which subsequent terms are obtained by multiplying the previous term by 2 and then subtracting 5.

This list is by no means exhaustive. There can be infinite ways to make a number series. It’s not possible to think or write about them here.

I got inspired to write this article after reading Series Chapter of the awesome book named Magical Book on Quicker Maths by M.Tyra. It can be a boon for any competitive exam aspirant. We’ll talk more about series in future. You can post any question related to number series as a comment below.

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Learn to test whether a given number is prime number or not

By looking at a number do you wonder whether it’s a prime number or not?
Is it always complicated for you to figure out the answer?

If the answer to above questions is yes, go ahead and learn this method of figuring out if a number is prime or not.

To test whether any number is a prime number or not, take an integer larger than the approximate square root of that number.

To quickly find the square root of any number, you can look at finding square root without calculator and Heron’s Method of finding roots.

Let say the square root of the said number is ‘x’. Test the divisibility of the given number by every prime number less than ‘x’. If it is not divisible by any of them then it is prime number; otherwise it is a composite number (other than prime).

Example 1: Is 349 a prime number?
The square root of 349 is approximately 19. The prime numbers less than 19 are 2, 3, 5, 7, 11, 13, 17.
Clearly, 349 is not divisible by any of them. Therefore, 349 is a prime number.

Example 2: Is 881 a prime number?
The square root of 881 is approximately 30. The prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
881 is not divisible by any of the above numbers. Therefore, 881 is a prime number.

If you know other ways of finding if a number is prime number or not, share it with all by posting a comment below.

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Calculate Square Root Quickly Without Calculator

In this post, we will learn how to find the square root of numbers which are not perfect squares. The answer we get using this quick calculation technique gives us an approximate answer. However, approximation becomes a necessity when we are attempting questions in a competitive exam, where time is short and options are given. Most of the time we’re not required to get exact answer.

Prerequisite to Use this Method

The prerequisite of using this method is you should remember the squares of as many numbers as possible. I would recommend that one should memorize square of numbers from 1 to 50. Later this can be extended t0 100.

It will be a wonderful idea to first learn the short cut method of finding the square of any number. Other awesome short cut method which you should consider knowing before moving forward is Herons method of finding roots.

Square Root of any number which is not a Perfect Square

Square Root = Sq. Root of Nearest Perfect Square + {difference of the given number from the nearest perfect square / 2 x (Sq. Root of Nearest Perfect Square)}

For example,

Short cut to find the Square Root of 47.
Square Root of 47
= 7 + (47-49) / 2 x 7 (since the perfect square closest to 47 is 49; we will take square root of 49 i.e. 7 for calculations)
= 7 – 2/2×7
=7 – 1/7
= 6.86 (approx).
The exact answer is 6.85565

Short cut to find the Square Root of 174
Square Root of 174
= 13 + (174-169) / 2 x 13 (since the perfect square closest to 174 is 169; we will take square root of 169 i.e. 13 for calculations)
= 13 + 5 / 26
= 13.19 (approx).
The exact answer is 13.1909

Short cut to find the Square Root of 650
Square Root of 650
= 25 + (650-625) / 2 x 25 (since the perfect square closest to 650 is 625; we will take square root of 625 i.e. 25 for calculations)
= 25 + 25/50
= 25.50 (approx).
The exact answer is 25.4950

This method has got a limitation. You can use it only till the point you remember the square
If you’ve questions related to the above shortcut method or anything else related to mathematics, please post on QM’s Q & A platform. For anything else you can comment below.

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Do You Know the Sum of All Positive Integers till Infinity, be Prepared for a Shock After Knowing the Answer

This is one of the most amusing mathematics trick pulled off by someone. If you understand basics of numbers, your jaw will drop at the output of this mathematics formulation. So here we go –

What do you think is the sum of all the integers up till infinity?

1+2+3+4+5……. so on to infinity=?

Anyone who know the meaning of infinity would quite safely say the answer is infinity, but practical implications and string theory in physics tells us otherwise.

The answer is  -1/12

If you think that’s impossible, check out the detailed explanation in this video:

This proof seem counter intuitive or to some it might even sound ridiculous. However, let me clarify this equation is completely accurate and not some hoax. It is used in theoretical physics.

Do you know of more such counter intuitive Mathematics Formulations?

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Shortcut to Find the Fourth Power of any Two Digit Number

In this article we will explore a shortcut to find the fourth power of any 2 digit number. The approach will be similar to the shortcut to find the cube of any 2 digit number. I strongly suggest that you should check that first. 

Generic Form of 4th Power of 2 Digit Numbers

The generic form of fourth power of any two digit number can be algebraically expressed as:

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

We will split the above result in 2 lines. We start with 4th power of 1st digit and then keep multiplying with ratio of 2nd digit: 1st digit = b/a. The remaining will go in the next line. This can be written as –

= a4 +  a3b  +  a2b2   +  ab3   + b4   

             3a3b + 5a2b2 + 3ab3

__________________________________  Adding the above 2 lines, we get original result.

= a4 + 4a3b + 6a2b2 + 4ab3 + b4

Example of Shortcut to Find the 4th power of 2 Digit Numbers

To try this shortcut, let us consider a simple example i.e. 124

Here the first digit is 1 and its fourth power is also 1. The ratio of 2nd to the 1st digit is 2 (Ratio = b/a)

The remaining 4 terms can be obtained by multiplying each of the previous terms as shown :

124 =  1      2       4      8     16

6      20    24

_____________________
=  1    8      24     32    16

=  1    8      24     32    16

=  1    8      24     33      6

=  1    8      27      3      6

=  2    0       7      3      6

= 20736

Let us try another example. We will find out the 4th power of 91.

914 = 6561            729             81            9           1

2187           405            27

____________________________________

6561        2916             486         36           1

Keeping single digit in each step and carrying the remaining digits and adding to number on the left, we get the answer as 68574961

Tip to use the shortcut: If we memorize the fourth power of all the single digits (0-9), it will come very handy for using this trick of calculating fourth power of any two digit numbers. Find below the table of fourth power of first ten natural numbers –

Number

Fourth Power

1

1

2

16

3

81

4

256

5

625

6

1296

7

2401

8

4096

9

6561

10

10000

Do you think this shortcut can be applied for calculating fourth power of a number?

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Learn Adding Numbers in Seconds

How would you add two numbers?

Let’s say I have to add 35 and 59.

35 + 59 would mean first add 5+9 = 14. Write 4 and carry over1. Then add this 1 to 3 and 5 which comes to 9. So our answer is 94. This is how we were taught in school. Now we will learn to do this at magical speed. However, before we delve into the quick addition method discussed in this post, check out how to avoid carryover while adding numbers.

We draw this line separator that divides the numbers into units, tens, hundreds and so on. We then add numbers in the same place or rather column together moving from left to right rather than right to left.

In our example we draw a line between 3 and 5 in 35 and 5 and 9 in 59. We add 3 and 5 first which is 8. Since it is at tens place this 8 is not 8 but 80. Now add 5 + 9 = 14. Now 14 = 10 +4, so we write 4 in the units place and carry over 1 or 10 to 3+5 or 30 + 50. And it becomes 80 + 10 +4.

3 / 5
5 / 9
——-
80
14
——-
94

At this stage you might be wondering what I am doing. Complicating things unnecessarily?

Well as of now you just added two digit numbers, once you get the hang of the concept you will be able to add 4 digit numbers also in a matter of seconds. Let me take some bigger numbers to demonstrate that.

Example: 986 + 37

98 / 6
3 / 7
——–
101
13
——-
1023

As you see 1010 + 10 +3 = 1023
To add 98 + 3 is a lot easier than adding 986 +37.
Now let us try adding two 3 digit numbers. Example: 575 + 789

57 / 5
78 / 9

Here we again break them i.e.shortcut for addition
5/7/5
7/8/9
———
12+
15+
14 this 4 is written in the units place without thinking

So it becomes
4
6 (5+1 i.e. 1 from 14 and 5 from 15)
13 (12+1 i.e. 12 and 1 from 15)
———–
1364

Now isn’t this method making much more sense? Similarly, if you wish to add consecutive numbers in a jiffy, here’s the shortcut trick to add consecutive numbers.

With some practice you will not even need to scribble this on paper.
Practice 20 to 30 additions and you will be doing it all mentally.  Don’t believe me. Give it a try and see it for yourself.

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How to Quickly Calculate Square of Three Digit Numbers?

Learn to Quickly Calculate Square of Any Three Digit Number

The method of squaring any 3 digit number is an extension of my last post on finding square of any two digit number. To understand and appreciate squaring of 3 digit numbers you should be well versed with shortcut of squaring any 2 digit number. Let us start learning this with the help of an example.

What is the square of 384?

Step 1: To begin with ignore the 3 of 384. You are left with only 84, a two digit number. Using the method of squaring 2 digit numbers, find the square of 84. We get the answer as

Square of 8  |   twice of 8 X 4   |  square of 4

64            64           16

7056 (consolidating the result obtained above)

Step 2: This step is new and different from what we’ve learned in the previous post for squaring 2 digit numbers. Watch carefully.

We have to multiply the first and last digits of our original number and double it. Essentially, that is multiplying together 3 and 4 and then doubling it. Hence we get 24.

Add this number directly to the two left hand digits of our number obtained from the first step.

7056

Add 24 to 70. 70+24=94. So 7056 gets converted to 9456.

Step 3: In the first step we left out the first digit of our number and squared the last two digits. Now we will forget about the unit’s digit 4 and square the first two digits i.e. 38 as before just omitting to square the last digit 8.

Square 38 as a regular 2 digit number, except that you omit the 8 squared.

Square of 3 | twice of 3 X 8

9       |   48

Step 4: Consolidating this with the result obtained in step 2,

9   |  48   |  9456

14        7       456

Hence the answer is 147456.

I’ve shared this method of squaring 3 digit numbers as an extension to the shortcut of squaring 2 digit numbers. Initially you might feel that the traditional method is quicker than having to memorize and execute these steps. However, this method can prove to be quicker than the useful one only if you master this technique with lot of practice.

Do you think this method will help you in reducing the time to calculate square of 3 digit numbers?

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How to Quickly Calculate Square of Any Two Digit Number?

In this post I’ve tried to improvise on the method of squaring presented in one of the earlier posts on Quickermaths.com itself. This trick of squaring any two digit number with ease is inspired by squaring techniques from the book – The Trachtenberg Speed System of Basic Mathematics

I would like to explain this method of squaring any 2 digits number with the help of an illustration.

What is the square of 32?

Step 1. In finding the last two digits of the answer, we shall find the square of the last digit of the number. Square the right-digit digit, which is 2 in this case. Hence we get 04

_ _04

Step 2. We shall now need to use the cross product. This is what we get when we multiply the two digits of the given number together. Multiply the two digits of the number together and double it: 3 times 2 is 6, doubled is 12: We write 12 as 2 and carry over 1 to the next step.

_ 24

Step 3: In finding the first two digits of the answer we shall still need to square the first digit of the number. That means we square the left hand figure of the number. Here square of 3 will be 9. Add 1 which is carried over from last step. Hence we get 9 + 1 =10

1024

That’s the answer.

This method can also be compared with another shortcut to find the square of any number posted by me on Quickermaths.com in the past.

Let’s try another example by squaring 64.

Square of 64 =     Square of 6 | double of cross product of both given digits 4 & 6| square of 4

Square of 64 =      36 |2x6x4 | 16

=      36 | 48 | 16

Collapsing the numbers

=      36 | 48 + 1 | 6

=      36 + 4 | 9 | 6

=      40 | 9 | 6

Hence the answer is 4096.

Have you come across any other squaring method like this?

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Multiplication Trick for Multiples of 11

If you know how to quickly multiply any number by 11 (click on the link to read further), the short cut multiplication method for 22, 33, etc. becomes easy to grasp. It’s an extension to the earlier method and you’ve seen earlier that there is no need to remember multiplication tables.

Multiplication by 11 is easy. Start from the right, add the two adjacent digits and keep on moving left. Since you can write only one digit in each step, if there is a carryover add it to the number obtained in the next step. So let’s begin to learn multiplication by multiples of 11.

Multiplication Trick for 22

For multiplication with 22, the rule is (number +next number)*2

Let us look at it step by step –
Step 1: For sake of simplicity, assume that there are two invisible 0 (zeroes) on both ends of the given number.
Say if the number is 786, assume it to be 0 7 8 6 0

Step 2:Start from the right, add the two adjacent digits and multiply by 2. Keep on moving left.
07860
Add the last zero to the digit in the ones column (6), and multiply by 2. Write the answer below the ones column.
Then add this 6 with digit on the left i.e. 8 and multiply by 2.
Next add 8 with 7 and multiply with 2.
Next add 7 with 0 and multiply by 2.
(0+7)*2     |    (7+8)*2  |    (8+6)*2   |   (6+0)*2
=   14   |   30   |  28   |  12

Step 3: Start from right most digit. Keep only the unit’s digit. Carryover and add the ten’s digit to the next number to the left. Doing this we get the answer as 17292.
Yes, job done. Quite simple, isn’t it?

Multiplication with Other Multiples Of 11

For multiplication with 33, the rule is (number +next number)*3
For multiplication with 44, the rule is (number +next number)*4
and so on….till
For multiplication with 99, the rule is (number +next number)*9. However, their is a simpler way of multiplication trick for 99

Multiplication Examples:
56789*22 = 0567890*22= (0+5)*2  |   (5+6)*2  |   (6+7)*2  |   (7+8)*2  |   (8+9)*2  |   (9+0)*2=1249358
123678*88 = 01236780*88 = (0+1)*8  |   (1+2)*8  |   (2+3)*8  |   (3+6)*8  |   (6+7)*8  |   (7+8)*8  |   (8+0)*8=10883664

Try it yourself. Share your experience with all by posting a comment below.

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