Integer Sequences


This lesson follows on from the previous one on Consecutive Integers.

Question 1

The solution to the above question is repeated with a more detailed explanation below.

There is a set of three consecutive even integers. When the smaller two in the set are added, they are the same as the difference between the largest number in the set tripled and 18.

Write an equation to represent the sentences above and solve it to find the three integers.

First, remind ourselves what consecutive even integers look like,

2, 4, 6, 8, ...

Note, the difference between consecutive integers for evens (and odds) is two, not one.

Let us write our definition for the consecutive even integers,

Let the first be: n

The second be: n + 2 (remember, going up in two)

The third be: n + 4

Now let's make our equation,

When the smaller two in the set are added, they are the same as the difference between the largest number in the set tripled and 18.

“The smaller two in the set are added”

That means the sum of the first two terms in the set, or,

n + (n +2)

That is the same as, “The difference between the largest number in the set tripled and 18”

The largest number in the set is n + 4.

So, we have the difference between triple (n + 4) and 18, this will look like, 3(n + 4) - 18,

We are told these two expressions “are the same” which we know means equal, so,

n + (n + 2) = 3(n + 4) - 18

Now we can solve,

Let’s first expand out our brackets,

n + n + 2 = 3(n) + 3(4) - 18

n + n + 2 = 3n + 12 - 18

2n + 2 = 3n - 6

Add 6 to both sides

2n + 2 + 6 = 3n - 6 + 6

2n + 8 = 3n

Subtract 2n from both sides

2n + 8 - 2n = 3n - 2n

8 = n

So, we have n= 8, which is our first term.

Our second term, n + 2 is 8 + 2, which is 10

Our third term is n + 4 which is 8 + 4, which is 12

So, our three consecutive even terms are, 8, 10, 12.

Question 2

The solution to the above question is repeated with a more detailed explanation below.

Three consecutive odd integers have the property that when the first term is subtracted from quadruple the second term, they are equal to ten greater than the last term.

Write an equation to represent the sentences above and solve it to find the three integers.

Remind ourselves again what consecutive odd integers look like, ... -1, 1, 3, 5, 7, 9, ...

Define our three terms as;

First is n

Second is n + 2

Third is n + 4

Note, even though they are odd numbers, they still go up by 2

Write our equation:

When the first term is subtracted from quadruple the second term,

Looks like,

When n is subtracted from quadruple (n + 2)

Which means,

4(n + 2) - n

And this is equal to ten greater than the last term

Which is 10 + (n + 4)

So,

4(n + 2) - n = 10 + (n + 4)

Expand out the brackets

4(n) + 4(2) - n = 10 + n + 4

4n + 8 - n = 14 + n

4n - n + 8 = 14 + n

3n + 8 = 14 + n

Subtract n from both sides

3n + 8 - n = 14 + n - n

2n + 8 = 14

Subtract 8 from both sides

2n + 8 - 8 = 14 - 8

2n = 6

Divide both sides by 2,

2n ÷ 2 = 6 ÷ 2

n = 3

So, our first term is 3,

Second is 3 + 2 = 5

Third is 3 + 4 = 7

Our three terms are 3, 5, 7.

Question 3

The solution to the above question is repeated with a more detailed explanation below.

The sum of four consecutive integers is -26. What are the four integers?

Write the definition for our four consecutive numbers. Remember, we are back to working with consecutive integers, so we go up by one.

Let the four consecutive integers be:

First: n

Second: n + 1

Third: n + 2

Fourth: n + 3,

Note: An easy mistake to make is being tricked into thinking the fourth term is n + 4 or the third is n + 3.

Don't be put off by seeing the negative number as the sum, this may hint that negative numbers are involved, but it does not mean to write the definition any differently. Following the correct methods will give the correct results.

We know the sum of these four integers will be - 26, so write the sum of these four integers using our four representations for the integers.

(n) + (n + 1) + (n + 2) + (n + 3) = -26

This might look new and unfamiliar but apply our learned methods as always.

Use the associative property and remove the brackets.

n + n + 1 + n + 2 + n + 3 = -26

n + n + n + n + 1 + 2 + 3 = -26

... [n + n + n + n = 4n] ...

4n + 1 + 2 + 3 = -26

4n + 6 = -26

Subtract 6 from both sides

4n + 6 - 6 = -26 - 6

4n = -32

Divide by 4 on both sides

4n ÷ 4 = -32 ÷ 4

n = -8,

So, our first term is - 8.

Our second is n + 1, [stay true to the +/-’s], -8 + 1 = -7

Our third is n + 2, -8 + 2 = -6

Our fourth is n + 3, -8 + 3 = -5

Our terms are -8, -7, -6, -5