Number series is a very interesting topic in reasoning that appears in most of the aptitude tests to evaluate your IQ. Number series questions are based on mathematical sequences that follow a logical rule or pattern and are based on elementary arithmetic concepts. Most of the time, you are given a sequence and you are asked to find the next number(s) in the sequence. You can also be asked to find a specific number in the middle of the series or to find the wrong term.

Let us now solve some tricky number series questions from different types of sequences to excel at this topic.

## Some common types of sequence in number series

**Arithmetic Sequence**

In an arithmetic sequence, the difference between any two consecutive terms is the same. For example, in the sequence, 4, 6, 8, 10, 12. The difference is 2 for any two consecutive numbers. That difference is known as common difference.

Let’ s solve some problems related to arithmetic sequence

**1) Find the next number in the following progression.**

-3/4, -1/2, -1/4, 0, ?

a) 1/2

b) -1/4

c) 1/4

d) 2

**Solution**

The correct answer is c.

This is an arithmetic progression with a common difference of 1/4. i.e. to get the next number in the progression add the previous by 1/4. Adding 1/4 to 0 gives the next term 1/4.

**2) Find the wrong element in the progression if any.**

66, 38, 12, -15, -42

a) 66

b) 38

c) -15

d) None

**Solution**

The right answer is b.

This an arithmetic progression with a common difference of -27. Adding -27 to the previous element gives the next element in the progression. When we add -27 to 66, the resulting number should be 39, not 38. Hence, 38 is the wrong element in this progression.

**Geometric Sequence**

A sequence in which the next term is obtained by multiplying a specific number or an order of numbers to the previous term is known as geometric sequence. In other words, there is a common ratio between any two consecutive terms. For example, in the sequence, 2, 4, 8, 16, 32. The common ratio is 2 i.e. 4/2 is 2, 8/4 is 2, and so on.

Let us now solve some number series questions of this type.

**1) What is the missing term in the sequence?**

1, 3/4, 9/16, ?, 81/256

a) 18/32

b) 15/20

c) 27/32

d) 27/64

**Solution**

The correct answer is d.

This is a geometric sequence with a common ratio of 3/4. The Next term is obtained by multiplying the previous term by 3/4. When 9/16 is multiplied by 3/4, the next number is 27/64.

**2) What is the wrong element in the following series?**

2.5, 1.25, 0.5, 0.3125, 0.15625

a) 2.5

b) 0.5

c) 0.15625

d) None

**Solution**

The right answer is b.

The above series is obtained by multiplying the previous term by 0.5 to get the next term. When we multiply 1.25 by 0.5. The next term should be 0.625, not 0.5. Hence, 0.5 is the wrong term in the series.

**Arithmetic–Geometric Sequence**

This type of progression is the combination of arithmetic and geometric progression.

Let us solve its questions to understand it.

**1) Find the next two elements in the following progression.**

4, 8, 24, 28, 84, 88, ?, ?

a) 264, 268

b) 336, 340

c) 124, 128

d) 94, 98

**Solution**

Option a is the correct option.

This is a arithmetic-geometric sequence in which the pattern is +4 and ×3 i.e. 4 + 4 = 8, 8 × 3 = 24 and so on.

84 + 4 = 88, 88 × 3 = 264, 264 + 4 = 268.

**2) Find the wrong term in the following progression.**

1/12, 1/4, -1/4, -3/4, 3/4, -15/4

a) 1/12

b) -15/4

c) 3/4

d) -1/4

**Solution**

The right option is c.

This series is the reverse of arithmetic-geometric progression. The pattern is to multiply by 3 first and then subtract 1/2.

For example, 1/12 × 3 = 1/4, 1/4 – 1/2 = -1/4, -1/4 × 3 = -3/4, -3/4 – 1/2 = -5/4.

Thus, the wrong term is 3/4.

**Mixed Number Series questions**

A Mixed series is the type of number series in which the series is formed by one or more patterns or rules. The series can also be created of any uncommon rules. Questions from this type of number series can be a bit tricky to solve because it can involve any unusual rule(s).

Let’s practice number series questions of this kind.

**1) What are the next two elements in this sequence: **

**7, 10.5, 21, 52.5, _, ?**

a) 83, 114.5

b) 157.5, 551.25

c) 93, 143.5

d) 127.5, 331.5

**Solution**

The correct answer is b.

The logic is 7 × 1.5 = 10.5,

10.5 × 2 = 21,

21 × 2.5 = 52.5,

52.5 × 3 = 157.5,

157.5 × 3.5 = 551.25

**2) Consider the series: 81, 54, 45, 42. Find the next number in the sequence.**

a) 31

b) 41

c) 27

d) 40.5

**Solution**

The correct answer is b.

The difference between two consecutive terms is divided by 3 each time. 81 – 54 = 27, 54 – 45 = 9, 45 – 42 = 3. The difference between the next term (missing term) and 42 should be 1. Therefore, the next term will be 41 i.e. 42 – 41 = 1.

**3) Find the wrong term in the sequence: 39, 40, 36, 45, 29, 55.**

a) 39

b) 36

c) 55

d) None

**Solution**

The right answer is c.

The sequence is obtained by alternatively adding and subtracting the square of positive integers.

39 + 1^{2} = 40,

40 – 2^{2} = 36,

36 + 3^{2} = 45,

45 – 4^{2} = 29,

29 + 5^{2} = 54.

Hence, the wrong term is 55. It should have been 54 according to the rule.

**Alternate series**

Alternate series is the type of series in which two or more series are combined into one. Each of the series in the alternate series is independent of one another.

**1) Find the next two elements in the sequence given below.**

2, 4, 3, 6, 5, 8, 7, 9, ?, ?

a) 13, 12

b) 10, 11

c) 11, 10

d) 12, 11

**Solution**

The correct answer is c.

This is a series consisting of two independent series i.e. series of first n prime and composite numbers. A composite number is followed by a prime number. 2, 3, 5, 7 and 4, 6, 8, 9. So, the next terms will be 11, 10 where 11 is a prime number succeeded by 10 which is a composite number.

**2) Find the missing terms.**

**2, 3, 8, 15, ?, ?, 128, 375.**

a) 32, 75

b) 32, 45

c) 88, 135

d) 64, 75

**Solution**

The right option is a.

The 1^{st} term is multiplied by 4 to get the 3^{rd} term and 2^{nd} term is multiplied by 5 to the 4^{th} term and so on. Therefore, there are two sequences: 2, 8, 32, 128 and 3, 15, 75, 375. Thus, the missing terms are 32 and 75.

**3) Find the wrong term in the sequence if any.**

**6, 7, 8, 7, 6, 9, 8, 5, 10, 8, 4, 11**

**Solution**

There are three parallel sequences. Adding 1 to 1^{st} term gives the 4^{th }term, subtracting 1 from 2^{nd} term gives the 5^{th }term, and adding 1 to 3^{rd} term gives the 6^{th} term. 6, 7, 8, 9 is the first sequence, 7, 6, 5, 4 is the second sequence and 8, 9, 10, 11 is the third sequence. Hence, the mistake is in the term of the first sequence. It should have been 9 instead of 8.

Also know how to find unit digit of number when it is raised to some power.

**Triangular Number Sequence questions**

A triangular number is a number if it can be represented in the form of equilateral triangle or that constitutes the equilateral triangle. Triangular number sequence is consists of these type of numbers. For example, 1, 3, 6, and so on. The next term could be found by [n(n+1)] / 2.

Let us solve its example to understand it.

**Find the wrong term in the series: 1, 3, 6, 10, 15, 22, 28, 36, 45.**

a) 10

b) 22

c) 36

d) None

**Solution**

The right answer is b.

This is a problem of triangular number sequence. 22 is the 6^{th} term in the sequence. Putting n=6 in the formula [n(n+1)]/2.

Next term = [6(7)]/2 = 42/2 = 21

Hence, the 6^{th} term should be 21, not 22. You can check other options by finding its position and putting in the formula to see whether the given term is right or not.

### Perfect square series questions

**What is the incorrect element in 484, 529, 576, 629, 676?**

a) 484

b) 629

c) 676

d) None

**Solution**

The correct answer is b.

This is a perfect square series i.e. 22^{2}, 23^{2}, 24^{2}, 25^{2}, 26^{2}. All of the other elements are right except 629. It should be 625 as 25^{2 }= 625.

### Perfect cube series

**Find the next number in 27, 125, 343, 729, ?**

**Solution**

This is a perfect cube sequence starting from 3 with a difference of 2 i.e. 3^{3}, 5^{3}, 7^{3}, 9^{3}. Thus, the next number will be 11^{3} = 1331.

**More Examples **

Let us solve some more number series questions.

**Problem**

**Find the missing elements in the series given below**

1, 1, 2, ?, 5, 8, ?, 21, 34, ?, 89

**Solution**

The series given above is a popular series known as the Fibonacci series. In this kind of number series questions, the next element is obtained by adding the previous two elements. For example, 1 + 1 = 2. Hence, the first missing element is 2 + 1 = 3, the second missing element is 8 + 5 = 13 and the last missing element is 21 + 34 = 55.

**Problem**

**What is the next element in the sequence: 3, 5, 9, 15, 23?**

**Solution**

This is one of the types of number series questions in which the difference between consecutive terms forms arithmetic or geometric sequence known as two-stage series. Here, 5 – 3 = 2, 9 – 5 = 4, 15 – 9 = 6, 23 – 15 = 8. The sequence of the differences is 2, 4, 6, 8 where the common difference is 2 and hence the next element of this arithmetic sequence will be 10. Therefore, 23 + 10 = 33 is our next element.

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