Tips to Solve Number Series

A number series is a collection of numbers that follow a pattern. Candidates must locate the missing or incorrect number in the given series. In some problems, one of the terms in the supplied series may be wrong, and candidates must figure out which term is faulty by recognizing the pattern involved in the series’ creation. Each question may follow a particular form of pattern or sequential arrangement of characters or digits, which applicants must detect using their common sense and thinking abilities.

The alternating series test is a method used in mathematical analysis to demonstrate that an alternating series is convergent when its terms decline in absolute value and approach zero in the limit. Gottfried Leibniz devised the test, which is also known as Leibniz’s test, Leibniz’s rule, or the Leibniz criterion. Some convergent alternating series may fail the first portion of the test because it is simply sufficient, not essential.

Various Types of Number Series

A series can be made in a variety of ways. Understanding these diverse techniques can aid us in spotting the pattern observed in the number series. So here are some examples of regular series.

There are several approaches for solving number series issues, including guessing the next number, i.e., which number will appear next, using rules such as addition, subtraction, and multiplication, or using other shortcuts.

Series Comprising of Perfect Squares:-

A Perfect Squares series is based on a specified arrangement of perfect squares of integers, and one of the numbers in this sort of series is usually missing, which you must find.

Xn= n² is the formula for solving number series problems.

Example: Find the missing number in the given series 441, 484, 529, 576, ?

Sol. The pattern follows the order

21² = 441

22² = 484

23² = 529

24² = 576

25² = 625

As a result the missing number is 625.

Perfect Cube Series

The series in this type consists of cubes of numbers in the same order, and you must locate the missing or odd cube number.

To answer number series questions, use the formula Xn=n³.

Example: Find the number 729, 6859, 24389, ?, 117649, 205379.

Sol. 9³ = 729

19³ = 6859

29³ = 24389

49³ = 117649

59³ = 205379

In this example, each cube number is added with 10 to become the next cube number. So, the missing one is 393 = 59319.

Rational Number Series

These are numbers that can be expressed as a fraction or quotient with an integer numerator and denominator.

Example: If 75 is divided into four parts proportionate to 3, 5, 8, and 9, find the least part.

Sol. Ration = 3:5:8:9

Sum of ratio terms = 25

The smallest part is (75 x 3/25) = 9.

Arithmetic Series

It is a mathematical series in which the integers differ by a defined amount. The following terms are obtained by either adding or removing a specified integer. Xn = x1 + (n – 1)d is the formula for solving number series issues. 

Example: Find the number 2, 4, 6, 8,?

Sol. The common difference between numbers is 2. So, the answer is 10.

Geometric Series

It’s a series in which each term is obtained by multiplying or dividing the previous number by a set number. 

Example: 3, 9, 27, 81,? Find the next number.

Ans. In this series each term is multiple of 3. So, the next one is 243.

Arithmetico-Geometric Series

The Arithmetico –Geometric series is a strange blend of Arithmetic and Geometric series, as the name suggests. The Geometric Sequence has differences in successive terms, which is an important aspect of this series.

Example: 4, 18, 60, 186,?

Ans. It follows the number series trick –

4, (4+2)x3, (18+2)x3, (60+2)x3

So the next one is (186+2)x3 = 564

Geometrico-Arithmetic Series

The Arithmetico–Geometric series is the inverse of the Arithmetico–Geometric series. The suggestive terminology differences in this series are in the Arithmetic Series.

Example: 2, 7, 17, ?, 77 get the missing one. 

Ans. It follows the number series trick – 2, (2×2)+3, (7×2)+3. So, the answer is (17×2)+3 = 37.

Tips to Solve Number Series

The following are some of the usual tips for calculating the next term:

• It’s possible that one number is a multiple of the other

• To get the second number, something may be added or subtracted from the first number, and so on

• To get the next number, divide the numbers

• To get the following one, the number can be squared

• To find the next number, take the square root of the numbers

• Even numbers are sometimes followed or preceded by odd numbers, and vice versa

• It is sometimes added to get the next number, while other times it is subtracted, split, or multiplied to get the following number

• An odd number is sometimes squared, the other multiplied three times, the fourth multiplied four times, and so on, before being added or subtracted

Also see: CAT Previous Year Papers

Conclusion

In Arithmetic, the concepts of sequence and series are crucial. A sequence is an itemised collection of elements that allows for any type of repetition. A series is the sum of all elements, whereas a sequence is an itemised collection of elements that allows for any type of repetition. One of the most common instances of sequence and series is mathematical progression. A sequence is a group of items or objects that are organised in a specific order. By adding up all of the terms in a series, it can be generalised. However, there must be a clear link between all of the sequence’s terms. Solving problems with formulas helps you understand the fundamentals. Individual terms in a sequence may appear more than once in different locations, comparable to sets. A sequence’s length is equal to the number of terms it contains, and it can be either finite or infinite.