Consecutive numbers are numbers that follow each other in order. They have a difference of 1 between every two numbers.
To solve word problems regarding consecutive integers, it is important to note :
Consecutive Integers :
If the first integer be x, then three consecutive integers be
x, x + 1, x + 2
Consecutive Even Integers :
If the first even integer be x, then three consecutive integers be
x, x + 2, x + 4
Consecutive Odd Integers :
If the first odd integer be x, then three consecutive integers be
x, x + 2, x + 4
Problem 1 :
Find two consecutive natural numbers whose product is 30.
Let x , (x + 1) be the two consecutive integers
Product of two consecutive integers = 30
x ⋅ (x + 1) = 30
x2 + 1x = 30
x2 + x - 30 = 0
(x + 6) (x - 5) = 0
x + 6 = 0
x = -6
x - 5 = 0
x = 5
Since those are positive integer we should not take x = -6, let us take 5 as the value of x.
Therefore two consecutive integers are 5 and 6.
Product of two positive integer is 30.
5 (6) = 30
30 = 30
Problem 2 :
There are three consecutive positive integers such that the sum of the square of first and the product of the other two is 154. Find the integers.
Let x, (x + 1) and (x + 2) be the first three consecutive integers
The sum of the squares of first and the product of the other two is 154
x2 + (x + 1) (x + 2) = 154
x2 + x2 + 2x + 1x + 2 = 154
2x2 + 3x + 2 = 154
2x2 + 3x + 2 - 154 = 0
2x2 + 3x - 152 = 0
(2x + 19) (x - 8) = 0
2x + 19 = 0
2x = -19
x = -19/2
x - 8 = 0
x = 8
Since it is positive integer, the value of x be 8.
Therefore three consecutive integers are 8, 9 and 10.
The sum of the square of first and the product of the other two is 154.
82 + (9) (10) = 154
64 + 90 = 154
154 = 154
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