Question

1. Find four consecutive even integers such that the sum of the first and third is 6 less than the largest

2. find four consecutive odd integers such that the largest equals 2 more than the sum of the other three


3. find two consecutive even integers such that twice the smaller is 26 less than three times the larger


4. find the consecutive integers such that twice the sum of the first and second is 27 more than three times the Third


5. what are three consecutive integers such that three times the smallest added to 7 is 13 less than four times the largest


6. if three times the first of three consecutive integers is subtracted from twice the second the result is 16 more than the third. find the integers


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


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


9. three consecutive multiples of 5 have a sum of 75 what are the numbers


I'm sorry these are a lot but I need help ASAP

Answer 1

These are indeed quite a lot of exercises, but a lot of them are almost identical - only small computations will change. So, I'm glad to help you with one exercise from every cathegory, but I encourage you to solve the others on your own.

Exercise 1

You can call four consecutive integers as

`x,\ x+1,\ x+2,\ x+3`

So, the sum of the first and third is
`x+(x+2) = 2x+2`

We want this quantity to be six less than the largest, i.e.
`(x+3)-6 = x-3`

So, the equality is

`2x+2 = x-3`

Subtract x from both sides:

`x+2 = -3`

Subtract 2 from both sides:

`x = -5`

So, the consecutive integers are

`-5,\ -4,\ -3,\ -2`

In fact, the sum of the first and third is
`-5-3 = -8`, which is indeed six less than the largest:
`-8 = -2-6`

Exercise 2

If you call the first odd number
`x`, the next consecutive odd numbers will be
`x,\ x+2,\ x+4,\ x+6`

In fact, we have to count skipping two's, because we only want odd integers. From here, you go on like exercise 1: you write the largest (which is x+6), and set it to be two more than the sum of the other three (x, x+2 and x+4)

Exercise 3

By the same logic of exercise 2, two consecutive even integers are
`x,\ x+2`, assuming that x is even.

So, you set the equation as usual: the smaller (which is x) is 26 less than three times the larger (which means 3(x+2)-26)

Exercise 4 to 8

These are all pretty identical to exercise 1: you start by listing three or four consecutive integers:

`x,\ x+1,\ x+2\quad\text{or}\quad x,\ x+1,\ x+2\ x+3`

and then you translate the request of each exercise accordingly. Remember that expressions like "three times the second number" means that you have to multiply:
`3(x+1)`, while expression like "six more than the first" or "thirteen less than the first" imply adding/subtracting:
`x+6` or
`x-13`.

Exercise 9

A multiple of 5 can be written as
`5k`, for some integer k.

So, three consecutive multiples of 5 are

`5k, 5(k+1), 5(k+2) = 5k, 5k+5, 5k+10`

We want these three numbers to have a sum of 75. So, we have

`5k, 5k+5, 5k+10 = 75 \iff 15k+15 = 75 \iff 15k = 60 \iff k = 4`

So, the three numbers are

`5k, 5(k+1), 5(k+2) = 5\cdot 4, 5\cdot 5, 5\cdot 6 = 20, 25, 30`