Solving the Reciprocal Equation of Consecutive Integers

Introduction

In this article, we will explore a problem involving the sum of the reciprocals of two consecutive integers. We will solve it using algebra, specifically by setting up and solving a quadratic equation. This problem is a classic example of how algebraic manipulation can be used to solve equations involving fractions.

The Problem

Let the two consecutive integers be n and n 1. According to the problem, the sum of their reciprocals is equal to -7/12. Mathematically, this can be written as:

Equation

1/n 1/(n 1) -7/12

Solving the Problem

To solve the equation, we first combine the fractions on the left-hand side:

Combining Fractions

(1/n) (1/(n 1)) (n (n 1)) / (n(n 1)) (2n 1) / (n(n 1))

Substituting this into the original equation gives us:

Rearranged Equation

(2n 1) / (n(n 1)) -7/12

Multiplying both sides by 12n(n 1) to clear the denominator:

Clearing the Denominators

12(2n 1) -7n(n 1)

Expanding and simplifying the equation:

Expanded Equation

24n 12 -7n^2 - 7n

Moving all terms to one side to form a standard quadratic equation:

Standard Quadratic Equation

7n^2 31n 12 0

Solving this quadratic equation using the quadratic formula n (-b ± √(b^2 - 4ac)) / (2a) where a 7, b 31, and c 12 gives:

Solving Using the Quadratic Formula

n (-31 ± √(31^2 - 4 × 7 × 12)) / (2 × 7) (-31 ± √(961 - 336)) / 14 (-31 ± √625) / 14 (-31 ± 25) / 14

This yields two solutions:

Solutions

n (-31 25) / 14 -31/14 -3

n (-31 - 25) / 14 -56/14 -4

Therefore, the consecutive integers are -4 and -3.

Verification

To verify, we substitute n -4 back into the original equation:

Verification for n -4

1/(-4) 1/(-3) -1/4 - 1/3 -3/12 - 4/12 -7/12

This confirms that the solution is correct.

Similarly, if n -3, we get:

Verification for n -3

1/(-3) 1/(-4) -1/3 - 1/4 -4/12 - 3/12 -7/12

This also confirms the solution.

Extension Problem

Consider the problem of finding any integers x and y such that the sum of their reciprocals equals -7/12. The equation is:

Extension Equation

1/x 1/y -7/12

Multiplying both sides by 12xy to clear the denominators gives:

Extension Expanded Equation

12x 12y -7xy

Multiplying both sides by 7 and adding 144 gives:

Extension Standard Equation

7x 7y 144 0

After simplifying and solving, we find the integer solutions are:

Extension Solutions

(x, y) (-2, -12), (-4, -3), (-3, -4)

The full solution set is therefore (-2, -12), (-4, -3), (-3, -4).

Conclusion

In this article, we solved the problem involving the reciprocal of consecutive integers and extended the solution to any integers. This problem demonstrates the power of algebra in solving complex equations and provides insights into number theory and the properties of integers.