Find the four consecutive integer numbers whose sum is 2

An algebraic expression is a combination of terms formed by the combination of variables as well as components. The terms in an algebraic expression are connected together using mathematical operations using addition or subtraction. The constants may be integers. The number of terms defines the algebraic expressions. There may be one or more variables in algebraic expressions. 

Consecutive Numbers

Consecutive Numbers are numbers that occur simultaneously on the number line. All such numbers have a difference of +1 or -1 between them. Some of the consecutive natural numbers are 1, 2, 3, 4, 5…  In the case of consecutive numbers, the difference between any predecessor-successor pair is fixed. 

Find the four consecutive integer numbers whose sum is 2

Solution: 

Approach 1: Using the AP formula 

Lets assume the numbers to x, x + 1, x + 2, and x + 3 respectively. 

If clearly notice, the numbers form an AP: 

Where the first term, a = x

Common difference, d = 1

Total number of terms, n = 4

Sum of n terms in an A.P = n/2[2a + (n – 1)d]

On substituting the values, 

2 = 4/2[2 × x + (4 – 1) × 1]

2 = 2[2x + 3]

1 = 2x + 3

On solving, 

2x = -2 

x = -1

Therefore, the first number is -1. 

The other numbers as, 

x + 1 = 0, x + 2 = 1, x + 3 = 2. 

Therefore, the numbers are -1, 0, 1, 2. The sum of these numbers is 2. 

Approach 2: Using the variable method

Lets assume the starting number to be x. 

Therefore, identifying the numbers, 

x, x + 1, x + 2 and x + 3. 

Now, 

Summation of these numbers is equivalent to 2. 

x + x + 1 + x + 2 + x + 3  = 2

On solving the equation, 

4x + 6 = 2

Equating the values, 

4x = -4

x = -1

Therefore, the value of x is equivalent to -1. 

Now, computing the numbers, 

x = -1

x + 1 = 0

x + 2 = 1 

x + 3 = 2

Therefore, the numbers are -1, 0, 1, 2. The sum of these numbers is 2. 

Sample Problems

Question 1: Find the three consecutive integers whose sum will be 657.

Solution: 

Assume the first number be x

Then the next two numbers will be x + 1, x + 2

Thus,

x + x +1 + x + 2 = 657

3x + 3 = 657

3x = 657 – 3

3x = 654

x = 654/3

x = 218

Therefore the integers are

x = 218

x + 1 = 219

x + 2 = 220

218 + 219 + 220 = 657

Question 2: If the measure of the lengths of the sides of the triangle is consecutive even numbers. Then find all the lengths of the triangle whose perimeter is 90 cm?

Solution:

Here as the sides are the consecutive even numbers, so add 2 to the previous side

Thus,

Length of side 1 = x

Length of side 2 = x + 2

Length of side 3 = x + 4

The perimeter of the triangle is sum of all the sides

Therefore,

Perimeter of the triangle = x + x + 2 + x + 4

90 = 3x + 6

3x = 90 – 6

x = 84/3

x = 28

Therefore,

Length of side 1 = x = 28

Length of side 2 = x + 2 = 30 

Length of side 3 = x + 4 = 32

Question 3: If the sum of three consecutive integers is -180. Find the largest integer?

Solution:

Assume the first integer be x

Thus,

The three integers will be

x, x + 1, x + 2

x + x + 1 + x + 2 = -180

3x + 3 = -180

3x = -180 – 3

3x = -183

x = -183/3

x = -61

x + 1 = -61 + 1 = -60

x + 2 = -61 + 2 = -59

Thus, the largest integer is -59

Therefore,

-59 + (-60) + (-61) = -180

Question 4: If the sum of five consecutive integers is 200. Find the second integer?

Solution: 

Assume the smallest integer be x

Therefore sum of all the integers will be

x + x + 1 + x + 2 + x + 3 + x + 4 = 200

5x + 10 = 200

5x = 200 – 10

5x = 190

x = 190/5

x = 38

Therefore,

The first integer will be 38

The second integer will be 38 + 1 = 39

The third integer will be 38 + 2 = 40

The fourth integer will be 38 + 3 = 41

The fifth integer will be 38 + 4 = 42

Thus,

38 + 39 + 40 + 41 + 42 = 200

The second integer is 39.

Question 5: Find the five consecutive odd integers following the digit -23.

Solution:

Here, to find 5 consecutive odd integers following -23

As it is known that of consecutive odd integers differ by 2

Thus, the integers will be like

x, x + 2, x + 4, x + 6, x + 8

If assumed x = -23

Then,

The first integer will be x = -23

The second integer will be x + 2 = -23 + 2 = -21

The third integer will be x + 4 = -23 + 4 = -19

The fourth integer will be x + 6 = -23 + 6 = -17

The fifth integer will be x + 8 = -23 + 8 = -15.