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MATH 60
UNIT 5.2
1
CONSECUTIVE INTEGERS
Reminder: Integers include all negative and positive whole numbers. Included are
(…, -2, -1, 0, 1, 2,…)
Consecutive integers are integers that follow each other in order such as 1, 2, 3, 4, 5, or -4, -3, -2, -1.
Example 1: Find 3 consecutive integers whose sum is 27.
First of all, we have to learn how to read the problem. There are clues in every math sentence. In our
problem we see that we are supposed to find 3 answers – all of which are integers.
We also know that the sum of our three answers must be 27. That means the three integers add up
to 27.
Notice that the word consecutive is left out. Consecutive means: one right after another, or in order. So if
our first answer was 8, the second answer would be the next integer in order – which is 9, and the third
answer would be 10.
In most problems we don’t know the answer before we start; however, once we assume that 8 is the first
answer, then the other answers, 9 and 10, naturally follow. But what if we don’t know the first answer?
We choose a variable, like n, to stand for our answer. Then the second answer will be one more than n, that
is, n+1. This is just like 9 is one more than 9. Similarly, the third answer will be n+2 (10 is 2 more than 8).
So let’s set up the problem:
We know that: (first) + (second) + (third) = sum
n
+
n+1
+ n+2 = 27
3n + 3
= 27
3n
n
So:
= 24
=8
n (the first number) = 8
n+1 (the second number) = 9
n+2 (the third number) = 10
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No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
MATH 60
UNIT 5.2
2
Example 2: Find two consecutive integers if twice the second integer, decreased by the first, is 5.
So,
let the first number be: n
let the second number be: n+1
So twice the second number 2 · (n+1)
We know that:
So:
twice the
2nd integer
decreased
by
2 · (n+1)
-
the
first
is
5
n
=
5
2n+2-n =
n+2
=
n
=
5
5
5
n ( the first integer ) = 3
n + 1 ( the second integer ) = 4
Examples of consecutive odd integers are 3, 5, 7, or -5, -7, -9. These are always two apart, so we can call
them n, n+2, n+4. Consecutive even integers (…, 4, 5, 6, 8, …) also differ by two, so we call them n, n+2,
n+4.
Example 3: Find 3 consecutive odd integers whose sum is -63.
1st integer = n
2nd integer = n+2
3rd integer = n+4
n+(n+2)+(n+4)
=
-63
3n+6
=
-63
3n
=
-69
n
=
-23
n = -23
n+2 = -21
n+4 = -19
Example 4: Find two consecutive even integers if three times the first plus twice the second is 44.
1st integer = n
3 times
2nd integer=n+2
the first
3n
plus
twice the
is
44
=
44
second
+
2 (n+2)
3n
+
2(n+2) =
3n+2n+4
1st integer:
2nd integer:
44
=
44
5n
=
44
n
n+2
=
=
8
10
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
MATH 60
UNIT 5.2
3
Problems:
Remember to include all integers that you’re asked to find.
1.
2.
3.
4.
5.
6.
7.
8.
Find two consecutive integers whose sum is 45.
Find 3 consecutive integers whose sum is -15.
Find 3 consecutive even integers whose sum is 120.
Find two even consecutive integers whose sum is -42.
Find two odd consecutive integers whose sum is 36.
Find two consecutive odd integers such that twice the first plus the second is 35.
Find three consecutive odd integers such that twice the first plus the last is 25.
Find three consecutive even integers such that three times the second, decreased by the first, is the
last.
9. Find three consecutive odd integers if four times the first is equal to the sum of the other two.
10. Find three consecutive even integers if twice the second integer, plus the first integer, is equal to
sixteen more than the third integer.
Answers:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
n + (n + 1) =45
n + (n +1) + (n + 2) = -15
n + (n + 2) + (n + 4) = 120
n + n + 2 = -42
n + n + 2 = 36
2n + n + 2 = 35
2n + n + 4 = 25
3 (n+2) – n = n + 4
4n = n + 2 + n + 4
2 (n+2) + n = 16 + n + 4
22, 23
-6, -5, -4
38, 40, 42
-22, -20
17, 19
11, 13
7, 9, 11
-2, 0, 2
3, 5, 7
8, 10, 12
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.