Consecutive Integers

Introduction

Consecutive integers are those integers that follow each other in a particular sequence or order. Natural numbers, whole numbers, rational & irrational numbers, real numbers & integers are some types of numbers. Integers are the set of whole positive & negative numbers with zero. ‘Integer’ is a Latin word which means whole or complete. This means fractional or decimal numbers are not included in integers.

• For example, 1, −5, 4, 9, −6. We can carry out basic arithmetic operations i.e addition, subtraction, multiplication & division on integers. Consecutive integers are those numbers that follow each other in a specific pattern or order. The difference in consecutive integers is always constant. These integers follow each other in ascending order. This concept is used while framing word problems.

Integers

• It is defined as, a set of the whole number with negative numbers & zero.

• Positive numbers, negative numbers & zero form a group of numbers known as

• They are denoted by Z.

• These numbers are the whole number they don’t have a fractional or decimal part.

• −5,0,8 − 6, 96,32 are some examples of integers.

• Integers are represented on the number line.

• Positive numbers are marked on the right side of the number line. Whereas negative numbers are marked on the left side of the number line.

• Zero is neither positive nor negative hence it is marked in the middle of the number line.

• This point is called the origin of the number line.

• The following figure represents an integer on the number line.

• These operations can be performed by using two ways, either by using algebraic rules & by using the number line.

Consecutive integers

• Consecutive integers are those numbers that follow each other in a particular sequence or pattern.

• The difference between each consecutive integer is constant i.e, fixed.

• For example, set of natural numbers, 1, 2, 3, 4, 5, 6,........., here we can see that difference between each term is one.

• If n is any number, then the sequence of numbers can be denoted as n+1, n+2, ……., n+m, where m represents the last number in the sequence.

• These numbers are in ascending order.

• The general formula of consecutive integers is n+1.

Consecutive positive & negative numbers

• To find the sequence of consecutive integers, the first integer must be positive or greater than zero to obtain a series of positive numbers.

• As consecutive integers are in ascending order, other numbers will, by default, be positive.

• This term is useful for counting.

• Whereas to find the sequence of consecutive negative integers, the first integer must be negative or less than zero to obtain a series of a negative number.

• As consecutive integers are in ascending order, other numbers will by default be negative.

Consecutive even & odd integers

Consecutive even integers

• Even numbers are multiple of 2. consecutive even integers are set of even integers.

• The difference between successive even integer is 2.

• Suppose 𝑧 is any integer then consecutive even integers can be written as 𝑧, 𝑧 + 2, 𝑧 + 4 etc.

• The general formula of consecutive even integer is 2n, where n is any integer.

Consecutive odd integers

• Odd numbers are not multiple of 2. Consecutive odd numbers are set of odd integers.

• The difference between successive odd integers is 2.

• Suppose z is any integer, then consecutive odd integers can be written as 𝑧, 𝑧 + 2, 𝑧 + 4 etc.

• The general formula of a consecutive odd integer is 2n+1, where n is any integer.

Product of two consecutive integers ( Always non- negative)

• Suppose m & n are any two consecutive integers, and the product of m & n is given as

$$\mathrm{Product\:=\:m\times\:n}$$

• For example, $\mathrm{2\times\:1\:=\:1}$

$\mathrm{--4\times\:-3\:=\:12}$

$\mathrm{-9\times\:-8\:=\:72}$

• From the above example, it can be seen that the product of consecutive numbers is multiple of 2.

• Hence the product of consecutive numbers is a multiple of 2 i.e even

• Also, the product of two integers is positive.

Solved examples

1) Find the missing number in the series 3, 6, 9, …., 15, 18, 21, 24?

Answer − The difference between successive numbers in a series is 3. The predecessor of the missing number is 9 & successor of the missing number is 15.

The missing number = predecessor + difference = 9 + 3 = 12.

2) If sum of three consecutive numbers is 51, then find three consecutive numbers?

Answer − Suppose the three consecutive numbers will 𝑛, 𝑛 + 1 & 𝑛 + 2.

The Sum of three consecutive numbers is 51

$$\mathrm{n\:+\:n\:+\:1\:+\:n\:2\:=\:51}$$

$$\mathrm{3n\:+\:3\:=\:51}$$

$$\mathrm{3n\:-\:51\:-\:3}$$

$$\mathrm{3n\:=\:48}$$

$$\mathrm{n\:=\:\frac{48}{3}}$$

$$\mathrm{n\:=\:16}$$

The first consecutive number is 16.

Second number $\mathrm{=\:n\:+\:1\:=\:16\:+\:1\:=\:17}$

Third number $\mathrm{=\:n\:+\:2\:=\:16\:+\:2\:=\:18}$

The three consecutive integers are 16, 17 & 18.

3) If the sum of four odd consecutive integers is 112, then find consecutive integers?

Answer − The difference between two consecutive odd numbers is 2. Suppose the first integer be 𝑛 second, third & fourth odd integer will be $\mathrm{n\:+\:2,n\:+\:4\&\:n\:+\:6}$ respectively.

Given that sum of four consecutive integers is 112.

$$\mathrm{n\:+\:n\:+\:2\:+\:n\:+\:4\:+\:n\:+\:6\:=\:112}$$

$$\mathrm{4n\:+\:12\:=\:112}$$

$$\mathrm{4n\:=\:112\:-\:12}$$

$$\mathrm{4n\:=\:100}$$

$$\mathrm{n\:=\:\frac{100}{4}}$$

$$\mathrm{n\:=\:25}$$

Other integers will be

$$\mathrm{n\:+\:2\:=\:25\:+\:2\:=\:27}$$

$$\mathrm{n\:+\:4\:=\:25\:+\:4\:=\:29}$$

$$\mathrm{n\:+\:6\:=\:25\:+\:6\:=\:31}$$

The four consecutive odd integers are 25, 27, 29 & 31.

Conclusion

• This tutorial covers the topic of consecutive integers in brief.

• In this tutorial, we have learned integers, consecutive integers & products of two integers with solved examples.

• Integers are the set of positive numbers, negative numbers & zero.

• They are the whole number, which doesn’t have the fractional or decimal part.

• Consecutive integers follow each other in a particular sequence or order.

• Also, we discussed consecutive positive & negative integers & consecutive even & odd integers.

• The difference between two successive odd & even integers is 2.

FAQs

1. State whether the following statement is true or false , The product of two consecutive natural numbers is always an even number?

True

2. What are non-consecutive integers?

It is a set of integers in which integers are not in a sequence. For example, 2, 5,10,21

3. Is the product of even no. of integers positive?

Yes. The product of an even number of integers is positive & product of an odd number of integers is negative.

4. What is an additive identity for integers?

The additive property states that when we add zero to any number, it gives the original number as an answer. Zero is an additive identity for an integer.

5. What is the sum of consecutive numbers?

The sum of consecutive numbers can be calculated by using the formula

$$\mathrm{\frac{n}{2}\times\:(first\:number\:+\:last\:number)}$$