Как найти два последовательных целых числа

Integers that follow each other in a regular counting sequence are known as consecutive integers. Because no numbers are missed in between when listing successive integers in a series, the difference between them is always constant i.e. difference between them is 1. We use consecutive integers, whenever we use them for numbers or counting items in a sequence. In other terms, consecutive integers are integers that follow each other in a fixed-difference sequence.

For Example: if we look at the natural number sequence 1,2,3,4,5,6,7,8… it’s showing a difference of 1 between all the numbers and that is fixed.

We can generate a list of consecutive even integers, consecutive odd integers, and several other combinations.

The main thing to know is that the difference between the consecutive numbers is constant, The numbers can be positive, negative, or zero, but they do not contain fractions or decimals because these are not part of integers.

Sequence of consecutive integer will be x , x+1 , x+2 , x+3 and so on ..

General term of consecutive integer = n

Consecutive Even Integers 

We know that even integers are multiples of 2. So, if we list the set of even numbers in ascending order, they may be expressed as -4, -2, 0, 2, 4, 6, 8, 10, and so on. We can see that the difference between each successive integer is 2. As a result, even consecutive integers have a difference of 2 between each predecessor and successor.

For example :10 – 8 Equals 2 and 8 – 6 = 2. 

If x is an even number, then the sequence can be written as :

 x, x + 2, x + 4, x + 6, and so on ..

General term of consecutive even integer = 2n

Consecutive Odd Integers 

Odd integers are the integers that are not be divisible by 2 and if we make up the sequence of odd numbers, then they can be written as 1,3,5,7,9 and so on…   

In consecutive odd integers also there is a difference of 2 which is constant in full sequence or The difference will be the same whatever the sequence is.

 For example :  9 – 7 = 2, and 7 – 5 = 2. 

So, if x is an odd integer, then the consecutive odd integers sequence will be as 

x, x + 2, x + 4, x + 6,…

General term of consecutive odd integer = 2n+1

Some Examples

Example 1: Find three consecutive integers after 64.

Solution:

Given:

Consecutive integer after 64 are

The three consecutive integers sequence after x will be x + 1, x + 2 and x + 3.

So, three consecutive integers after 64 are = 64 + 1, 64 + 2 and 64 + 3

                                                               = 65,66,67 

Therefore, three consecutive integer after 64 will be 65,66,67 

Example 2: Find 5 consecutive even integers of -10?

Solution:

 Given: Consecutive integer after – 10 , as it is  an even negative integer .

Therefore , Let – 10 be x . 

So the next five consecutive even integers will be  x + 2, x + 4, x + 6, x+8 , x+10 and so on…

                                                                =  x + 2, x + 4, x + 6, x+8 , x+10

                                                                =  (-10 + 2 ) , (-10 + 4 ) , (-10  + 6) , (-10 +8) , ( -10 +10) 

                                                                =  –8 , -6 , -4 , -2 , 0

Therefore, the five even consecutive integers will be have -10 , -8 , -6 , -4 , -2 , 0.

Example 3: Find the smallest number if the sum of four consecutive odd integers is 288? 

Solution:

Given: sum of four consecutive odd integers are 288

So , The four consecutive odd integers sequence will be  x, x+2, x+4 and x+6 

According to the given sum ,

                       x + x + 2 + x + 4 + x + 6 = 288

                                                   4x + 12 = 288

                                                           4x = 288 – 12

                                                           4x = 276

                                                             x = 276/4

                                                             x = 69

Hence, the smallest number will be 69.

Example 4: If the sum of four consecutive integers is 302, then what is the product of the first and the third integer?

Solution:

Lets assume four consecutive integers is   x , x + 1 , x + 2 , x+3 

Given: Sum of the four consecutive integers are 302

therefore ,

x + x + 1 + x + 2 + x + 3 = 302

                              4x + 6 = 302

                                     4x = 302 – 6

                                     4x = 296

                                       x = 296/4

                                       x = 74

so the first integer is       x = 74

and third integer  is  x + 2 = 76

Product of the first and third integer = 74 × 76 = 5624

Example 5: Find two consecutive positive integers sum of whose squares is 265?

Solution:

Given: Two consecutive positive integers sum of whose squares is 265

Lets Assume one integer be x, then the other integer is x + 1, 

So , We  have x² + (x + 1)² = 265

⇒            x² + x² + 1 + 2x  = 265               —           [BY Using algebraic identity (a + b)² = a² + 2ab + b²]

⇒                   2x² + 2x + 1 = 265

⇒          2x² + 2x + 1 – 265 = 0

⇒                 2x² + 2x – 264 = 0

⇒                       x² + x – 132 = 0            

⇒              x² + 12x – 11x – 132 = 0

⇒           x(x + 12) – 11 (x + 12) = 0

⇒                   (x +12) (x -11) = 0

⇒                      x = -12 or x = 11

Since, we need a positive integer, 

therefore x = -12 is rejected. So, x = 11.

Then, x + 1 

      =  11+1

      = 12 

hence first two positive consecutive integers are 11, 12.

Найдите два последовательных целых числа, сумма квадратов которых равна 1.