Consecutive Integers Sequences


This lesson will build on our word problem solving skills and our understanding of number sequences.

Let's quickly remember our Integer definition:

All negative and positive whole numbers i.e. ... -2, - 1, 0, 1, 2 ...

Consecutive Integers are any list or sequence of integers only separated by 1. It does not matter how long. For example, 1, 2, 3 or 99, 100, 101 or -5, -4, -3, -2.

Consecutive odds and evens is the same principle but these are separated by 2.

Consecutive Evens:

2, 4, 6 or -88, -86, - 84

Consecutive Odds:

-3, -1, 1, 3

Question 1

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

There are two consecutive integers whose sum is the difference between 17 and double the smaller integer. Use this information in the following 4-part task:

i) Try 3 and 4 to see if they are the two consecutive integers.

ii) Define the two consecutive integers using let statements.

iii) Make an equation based on our question.

iv) Solve the equation to give us our two integers.

i) Try 3 and 4 to see if they are the two consecutive integers.

So, the sum of 3 and 4, let’s see if it is the same as the difference between 17 and double the smaller integer,

Sum of 3 and 4 is 3 + 4 = 7,

The difference between 17 and double the smaller integer (3) is

17 - 2(3) = 17 - 6 = 11,

We can see the integers are not 3 and 4. However, the purpose of this task is to help you see how you will solve future ones.


ii) Define the two consecutive integers using let statements.

This might at first seem difficult, but we know that consecutive integers are just one more than the one before (i.e. 1, 2 or 7, 8), so,

Let our first integer be:

n

Let our second integer be:

n + 1


iii) Make an equation based on our question.

We need to make an equation out of “There are two consecutive integers whose sum is the difference between 17 and double the smaller integer”

We now know our two integers can be written as n & n + 1, so we can write the sum of them as,

n + (n + 1)

This is equal to the difference between 17 and double the smaller integer,

Which is,

17 - double the smaller integer,

17 - 2 (the smaller integer OR the first integer),

17 - 2(n)

So,

n + (n + 1) = 17 - 2n is our equation.


iv) Solve the equation to give us our two integers.

Using the methods we are getting familiar with (Associative, Commutative, Distributive Properties and Additive and Multiplicative Equalities):

n + (n + 1) = 17 - 2n

n + n + 1 = 17 - 2n

... [n + n = 2n] ...

2n + 1 = 17 - 2n

Subtract 1 from both sides,

2n + 1 - 1 = 17 - 2n - 1

2n = 17 - 1 - 2n

2n = 16 - 2n

Add 2n to both sides,

(2n) + 2n = (16 - 2n) + 2n

4n = 16

Divide both sides by 4,

(4n) ÷ 4 = 16 ÷ 4

n = 4,

Remember n = 4 is not our answer alone!

n = 4 is our first integer, so our second integer (n + 1) is (4 + 1) = 5

Our two consecutive integers are 4 and 5.