The questions asked under this test are based on number or letter. Through this test, the candidate’s ability to calculate faster is tested. it is also examined how quickly you determine the relations between them, depending on their status between different letters or digits. The questions asked under this test can be categorized into the following sections.

Under this, a series of Numbers are given. This series is based on addition, subtraction, multiplication, division, square, square root, cube, cube root, etc. To solve these questions, it is necessary to focus on the key points given below.

** (i) **If the value of the given series Numbers is going to increase in general, then definitely the series belongs to the addition.

** (ii)** If the value of the given series Numbers is going to decrease in general, then there is definitely a subtraction mechanism.

** (iii)** If the number of given series is increasing with a lot of intensity, then surely there is a work of multiplication or square, it is also possible that addition and subtraction or addition or subtraction is happening together.

** (iv)** If the numeric value of the chain is decreasing with intensity, then the possibly there is division function. And addition or subtraction is also possible.

** (v) **If the series increases with intensity first and later decreases, then there are multiplication and division actions being followed one by one.

** (vi)** If the value of digits in the chain first increases then decreases but at very least difference, then the function of addition and subtraction is done after one another.

** Normally two types of questions are asked under the number series **

**(a) Complete the given series: **In this given sequence order, a particular place is left blank or denoted by a question mark (?), Then you have to select the appropriate number from given options to be placed in place of the question mark (?) by observing the sequence.

Now observe the examples given for explanation of the above facts carefully

**Example 1. **Which of the following options will be placed in place of the question mark (?) in the following series?

16, 23, 31, 40, 50, 61,?

** (a) 81 (b) 77 (c) 83 (d) 81**

** Solution:** (a) Carefully observing the above series, we find that the series is increasing in order of +7, +8, +9, + 10. Thus, it can be expressed as follows.

Therefore, the appropriate number of in place of question mark (?) Will be 73.

**Example 2. **In the series of the following points, which of the following options will be placed in place of the question mark (?)?

12, 6, 6, 9,? , 45, 135

(a) 18 (b) 24 (c) 28 (d) 32

Solution: After careful observation of the above series, we find that the series is in increasing order.

So the appropriate number of question marks (?) Will be ’18 ‘.

**Example 3. **What will come in place of the question mark (?) in the following series?

144, 100, 64,?, 16, 4

(a) 49 (b) 42 (c) 36 (d) 51

Solution: (c) After carefully observing the above series we find that the given series is based on the class which is as follows

So the number in place of the question mark (?) Will be 36.

**Example 4. **Which number will come in place of the question mark (?) in the following series?

2, 5, 1441, 122, 365, 2

(a) 1000 (b) 1094

(c) 1059 (d) 1029

Solution: (b) After carefully observing the series, we find that each next post is increasing with more difference, so we can guess that the action in it will be of multiplication. Addition or subtraction is also possible. So we find that series is increasing in the order of × 3 – 1, × 3 – 1,

So the intended number will be ‘1094’ in place of the question mark (?).

(b) Finding the wrong Term in a given series

In the series sequence given below, a wrong digit is added instead of the number coming to a particular place, you have to find the wrong term used in the series by identifying the order given to the candidates. For this, the candidates should first know that the terms in the category are changing according to which rule, then it should be known that according to the rule, which term is not changing, that is the wrong post.

Now, observe some examples carefully for clarification of the above facts.

**Example 1. **Only one term in the following numbers category is incorrect, find out the wrong term.

3, 4, 7, 11, 20, 29, 47, 76

(a) 7 (b) 11 (c) 20 (d) 47

Solution: (c) After observing the above-mentioned category, it is known that the third term of the series is the sum of the first and the second term, and the fourth term is the sum of the second and third terms. Thus, each term is equal to the sum of its previous two posts, but this rule does not apply on the term 20 used in the series, because before 20, two terms are there 7 and 11, whose sum is 18. So the wrong term is ’20.’

it Can be expressed as follows

That is 3+ 4 = 7, 4 +7 = 11, 7 + 11 = 18, 11 + 18 = 29, 18 + 29 = 47, 29 + 47 = 76

Hence the number ’20 ‘in the series is an inappropriate number because the number of 20 should be ’18’.

**Example 2. **What is the number in the series that is inappropriate in the series?

5, 3, 6, 10, 9, 12, 17, 15, 18, 23

(a) 6 (b) 10 (c) 9 (d) 12

Solution: (b) After carefully observing the above series we find that given series Decreasing and increasing in the order of – 2, + 3, + 5, – 2, + 3, + 5,.

which can be expressed as follows

So the above series should have term ’11” after 6. So there is an inappropriate number 10 in the series.

**Example 3. **One post in the following numbers category is incorrect, you have to find the wrong post, and also find the correct term in place of that wrong post from the given options?

105, 114, 127, 138, 153, 170, 189

(a) 114 (b) 125 (c) 138 (d) 170

Solution: (b) After careful observation of the above series, it is known that the difference of two successive terms is 9, 13, 15, 17 and 19, respectively. In the given series, the difference between two terms is to 2 except for the second and the third, if 125 is placed at the place of 127, then the gradual terms of the class will be 9, 11, 13, 15, 17 and 19 respectively. Which are growing uniformly. So 127 is wrong and it should be 125 in its place. It can also be expressed as follows

So wrong post = 127 and correct position = 125

**Example 4. **What is the number in the following series that is inappropriate?

1, 4, 3, 16, 5, 32, 7, 64

(a) 4 (b) 3 (c) 16 (d) 32

** Solution: **(a) The given series is divided into two sections. In this, the number of each odd places is increasing + 2 + 2 + 2, respectively, while the number of places is increasing in the sequence of × 2 × 2 × 2, … In the given series, there should be number 8 in place of number 4, it has been expressed as

So incompatible numbers = [4]

**⇒ Apart from the rules of Progression,** questions can also be asked which are usually of two types:

is called a category in which the difference of two consecutive terms is same, the number received after subtracting a term from its next term is known as common Difference.

So if the first position of the Arithmetic Progression(A) is **a** and the common difference is **d**, then the Arithematic Series will be

a, (a + d), (a +2d), (a + 3d), ……

Therefore, the nth grade of the Arithmetic Progression is T = a + (n-1) d (where a is the first term and d is the common difference)

** Example 1. **Category 3, 5, 7, 9,. What will be at the 10th rank?

Solution: 10th position T_{10} = a + (10-1)d

= 3 + (10-1)2 = 21

(Since the first term here a = 3 and the derivative d = 5 – 3 = 7 – 5 = 9 – 7, i.e. 2)

** Example 2. **Category 357, 363, 369, 375, …. Which of the following is the 12th term?

(a) 423 (b) 411 (c) 442 (d) 449

Solution: (a) 12th term T_{12} = a + (12 – 1) d

= 357 + 11 * 6 = 357 + 66 = 423

** Example 3. **Find out the number of posts if the first step of an AP is 5, the common differeence is 3 and the last position is 80.

(a) 24 (b) 23 (c) 26 (d) 29

Solution: (c) here a = 5, d = 3, T_{n} = 80, n =?

T_{n} = a + (n-1) d ** ⇒ **80 = 5 + (n-1) 3

⇒ (n -1) = (80 – 5) / 3 = 25 ⇒ n = 25 + 1 = 26

Hence the number of terms n = 26

A category in which the ratio of two consecutive terms is the same. This ratio is called a common ratio of GP. The common ratio of the GP is obtained by dividing the post from its previous post.

If the first term of a ratio category is **a** and common ration is **r**, then the GP’s nth term T_{n} = a.r ^{ n-1 }

** Example 1. **Which of the following is the 6th term of class 3, 9, 27, 81, ..?

(a) 243 (b) 729 (c) 1681 (d) 1747

Solution: (b) The first position here is a = 3 and the common ratio is = 3

6th position T_{6} = ar ^{ n-1 }= 3.3 ^{ 6-1} = 3 × 3^{5} = 3 × 243 = 729

Common ratio r = 9/3 = 27/9 = 81/27 that is 3

** Example 2 .** Series 7, 14, 28,. What will be the 10th post?

(a) 3584 (b) 2684 (c) 2736 (d) 3216

Solution: (a) Here the first term a = 7 and the common ratio r = 2

10th term T_{10} = a.r ^{ n-1 }= 7.2 ^{ 10-1}

= 7 × 2^{9} = 7 × 512 = 3584

Under this series, a series of letters related to the English alphabet is given, in which one or two letters are lost or the location is denoted by the question mark (?), you have to find the correct letter to replace the mark (?). For this, the candidates have to carefully observe the given series and the series is changing according to which rule and according to this changed rule, what is the appropriate term in place of the question mark (?).

To easily solve the questions related to this series, it is necessary to remember the numeric values of the alphabet like A = 1, B = 2, C = 3, similarly Z = 26.

Now, let’s carefully examine some examples and their interpretive solution for the explanation of the above facts.

**Type 1 **

**⇒ instructions (example 1-4)** Each question given below is given a series of letters. One or two letters have been omitted in these series and the question mark (?) Has been shown in place of them. Carefully study the given series, find out one of the four options given below, which is suitable in place of the question mark (?) In the series.

** 1. J K M P T ?**

(a) X (b) W (c) Y (d) None of these

** 2. Z L X J V H T F? ?**

(a) SE (b) RD (c) QD (d) RE

** 3. M P N Q O R? ? Q T**

(a) SU (b) PS (c) QR (d) VO

** 4. AZX BVT CR? ?**

(a) PD (b) EQ (c) QE (d) PS

** Solution: 1. **(c) After carefully observing given alphabet sequence, we find that the letters used in the range are + 1, + 2 + 3 + 4, respectively, from their alphabetical order.

So the appropriate letter is “Y” in place of the question mark (?) .

** 2. **(b) Upon careful observation of the series, we find that the letter chain is divided into two sections. Under the first section, the alphabet used in the odd location in the series is in alphabetical order according to the numerical value decreasing by 2, then under the second section, the letters used in place of the even numbers are also in alphabetical serial value, decreasing by 2. It can be displayed in the following form.

So the appropriate letter in place of the question mark (?) will be RD.

** 3. **(b) On careful observation of the given letter series, we find that the series is divided into two sections, the letter on the odd position Are growing in the order of + 1, + 1, + 1 , the letter on the even position Are growing in the order of + 1, + 1, + 1 , it can be expressed as follows

So the appropriate letter PS will be in place of the question mark (?).

** 4. **(a) The given series is organized in the following order

So the correct letter to replace the question mark (?) Is PD.

**Type 2 **

**Example 1**. Which of the following options in the series of the following characters comes in place of the question mark (?)?

BXF, DVI, FTL, HRO?

(a) JOL (b) KPM (c) KPL (d) JPR

** Solution:** (d) After carefully observing the given letter series, we find that the first letter of each group of the letter series increasing In the sequential order + 2 + 2 + 2, …….. , the second letter of each group decreasing In the sequential order – 2, – 2, – 2, ……… and the third letter of each group is organized in increasing order + 3 + 3 + 3. , which is expressed as follows

So the appropriate group of letters coming in place of the question mark (?) Will be ‘JPR’.

**Example 2.** Which of the following options in the series of the following letters comes in place of the question mark (?)?

EDBA, KJHG, QPNM?

(a) ZXUV (b) WVTS (c) KIGH (d) QOMK

** Solution:** (b) After carefully observing the above letter series, we find that the first letter used in each group of the series is + 6 + 6 + 6, rather than its alphabet serial value. Similarly, the second letter of each group, the third letter, and the fourth letter are also being changed in this order. It can be expressed as follows

So the appropriate letter group “WVTS” will be in place of the question mark (?).

**Type 3 **

**⇒ instructions (example 1-3)** Each question given below has a combined series of letters and digits. In this series, a few digits and letters have been omitted and the question mark is displayed in its place (?), By carefully studying the given series, find out one of the four options given below in the series, which can be fit in place of the question mark (?) In the chain.

1. L7C N9F P12I R16L?

(a) T21O (b) S20O (c) S21P (d) U21Ο

2. KM5 IP8 GS11 EV14?

(a) BY17 (b) CZ17 (c) BX17 (d) None of these

3. B8Y9 D7W7 F8U8 H5S6 J4O4?

(a) L5O6 (b) L5O7 (c) L5P5 (d) L5O5

** Solution: 1. **(a) Carefully observing the series, we find that the first letter of each group is 2 places ahead of the first letter of its previous group, then the middle number in each group Number is increased by + 2 + 3 + 4,. And the last letter of each group is 3 places ahead of the last letter of its previous group, it has been expressed in the following form

So instead of the question mark (?), There will be appropriate digits T21O.

**2. **(d) After carefully observing the given series, we find that the first letter of each group is based on its alphabet serial number – 2 – 2 – 2 from the first letter of the previous group. In the same way, the middle letter of each group is increasing in order of + 3 + 3 + 3 between each group. In addition to this, there is a digit at the end of each group, which is increasing in order of + 3 + 3 + 3.. The above fact is expressed in the following form

Hence CY17 will be there in place of (?).

** 3. **(d) Having a careful overview of the given series, we find that the first letter of each group used in the series increased by + 2 + 2 + 2. Then in the second place of each group and in the fourth position, In each 3 Groups First it increases by 1 point Then decreases by 1 respectively. Then 3rd point in each group is decreasing in the sequence of – 2 – 2 – 2., it can be understood as the following form

So ? = L5O5

**Type 4 **

**⇒ instructions (example 1-4)** Each of the following questions is given a series of letters. Some characters have been abolished in these series and they are given in the sequence given below in the sequence of the series as it should be in the series. By carefully studying the given series, find out one of the four options given below, Appropriate in place of missing letters.

** 1.** ab_baabc_aabcb_abcb_

(a) bcaa (b) cbaa (c) abca (d) aacb

2. a_bbaa_baa_baab aab

(a) abab (b) baba (c) abbb (d) babb

3. aaa_bb_aab_baaa_bb

(a) abab (b) bbaa (c) aabb (d) babb

4. _sr_tr_SrS_r_SrSt_

(a) ttsSrr (b) tsrtSr (c) Strtrs (d) tstttr

** Solution: 1. **(b) At the beginning of the series given to solve such questions, we see that the letter ‘b’ is used on either side of the empty space and in that series The ‘c’ letter is used in between two ‘b’, so at the beginning of the series, the letter ‘c’ will be used in the space between two ‘b’, thus the chain formed

ab c baabc_aabcb_abcb_ On the side of the next empty space in the reclaimed series, we see that the letter C and A have been used and behind the back of the chain the letter b is used between c and a, so the series will be lower

ab __ c__ baabc b aabcb_abcb_

Again the letter b and a are used on both sides of the next empty space, and the letter ‘a’ is used between b and a right behind the chain. Hence the series will be low

ab c baabc b aabcb b aabcb a abcb_

Again the ‘abcb’ letter is used behind the next empty space and just behind the chain we see that the letter ‘a’ is used after abcb, this series type chain will be lower ab c ba / abc b a / abcb a / abcb a

Therefore, the intended reply will be ‘cbaa’ according to the option.

** 2. **(c) The given series will be organized as follows

a a bb / aa b b / aa b b / aab b / aab

Therefore, the answer to the option will be ‘abbb’.

** 3. **(d) The given series will be organized as follows

aaa / b bb / a aa / b b b / aaa / b bb

Therefore, according to the alternative, the intended answer will be ‘babb’.

** 4. **(d) The given series will be organized as follows

t sr / s tr / t sr / s t r / t sr / st r

Therefore, the desired answer according to the option will be ‘tstttr’.

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