__Problem 1:__
Find two consecutive integers whose sum is equal 129.

__Solution to Problem 1:__

Let x and x + 1 (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 129 to write the equation

x + (x + 1) = 129

Solve for x to obtain

x = 64

The two numbers are

x = 64 and x + 1 = 65

We can see that the sum of the two numbers is 129.

__Problem 2:__
Find three consecutive integers whose sum is equal to 366.

__Solution to Problem 2:__

Let the three numbers be x, x + 1 and x + 2. their sum is equal to 366, hence

x + (x + 1) + (x + 2) = 366

Solve for x and find the three numbers

x = 121 , x + 1 = 122 and x + 2 = 123

__Problem 3:__
The sum of three consecutive even integers is equal to 84. Find the numbers.

__Solution to Problem 3:__

The difference between two even integers is equal to 2. let x, x + 2 and x + 4 be the three numbers. Their sum is equal to 84, hence

x + (x + 2) + (x + 4) = 84

Solve for x and find the three numbers

x = 26 , x + 2 = 28 and x + 4 = 30

The three numbers are even. Check that their sum is equal to 84.

__Problem 4:__
The sum on an odd integer and twice its consecutive is equal to equal to 3757. Find the number.

__Solution to Problem 4:__

The difference between two odd integers is equal to 2. let x be an odd integer and x + 2 be its consecutive. The sum of x and twice its consecutive is equal to 3757 gives an equation of the form

x + 2(x + 2) = 3757

Solve for x

x = 1251

Check that the sum of 1251 and 2(1251 + 2) is equal to 3757.

__Problem 5:__
The sum of the first and third of three consecutive odd integers is 131 less than three times the second integer. Find the three integers.

__Solution to Problem 5:__

Let x, x + 2 and x + 4 be three integers. The sum of the first x and third x + 4 is given by

x + (x + 4)

131 less than three times the second 3(x + 2) is given by

3(x + 2) - 131

"The sum of the first and third is 131 less than three times the second" gives

x + (x + 4) = 3(x + 2) - 131

Solve for x and find all three numbers

x = 129 , x + 2 = 131 , x + 4 = 133

As an exercise, check that the sum of the first and third is 131 less than three times

__Problem 6:__
The product of two consecutive odd integers is equal to 675. Find the two integers.

Let x, x + 2 be the two integers. Their product is equak to 144

x (x + 2) = 675

Expand to obtain a quadratic equation.

x^{ 2} + 2 x - 675 = 0

Solve for x to obtain two solutions

x = 25 or x = -27

if x = 25 then x + 2 = 27

if x = -27 then x + 2 = -25

We have two solutions. The two numbers are either

25 and 27

or

-27 and -25

Check that in both cases the product is equal to 675.
__Problem 7:__
Find four consecutive even integers so that the sum of the first two added to twice the sum of the last two is equal to 742.

__Solution to Problem 7:__

Let x, x + 2, x + 4 and x + 6 be the four integers. The sum of the first two

x + (x + 2)

twice the sum of the last two is written as

2 ((x + 4) + (x + 6)) = 4 x + 20

sum of the first two added to twice the sum of the last two is equal to 742 is written as

x + (x + 2) + 4 x + 20 = 742

Solve for x and find all four numbers

x = 120 , x + 2 = 122 , x + 4 = 124 , x + 6 = 126

As an exrcise, check that the sum of the first two added to twice the sum of the last two is equal to 742

__Problem 8:__
When the smallest of three consecutive odd integers is added to four times the largest, it produces a result 729 more than four times the middle integer. Find the numbers and check your answer.

__Solution to Problem 8:__

Let x, x + 2 and x + 4 be the three integers. "The smallest added to four times the largest is written as follows"

x + 4 (x + 4)

"729 more than four times the middle integer" is written as follows

729 + 4 (x + 2)

"When the smallest is added to four times the largest, it produces a result 729 more than four times the middle" is written as follows

x + 4 (x + 4) = 729 + 4 (x + 2)

Solve for x and find all three numbers

x + 4 x + 16 = 729 + 4 x + 8

x = 721

x + 2 = 723

x + 4 = 725

Check: the smallest is added to four times the largest

721 + 4 * 725 = 3621

four times the middle

4 × 723 = 2892

3621 is more than 2892 by

3621 - 2892 = 729

The answer to the problem is correct.

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