Exploring the Difference Between Squares of Consecutive Integers

Solving mathematical problems can sometimes be as simple as breaking them down into smaller, more manageable parts. One intriguing problem involves understanding the relationship between the squares of two consecutive integers and their difference. Let's dive into the solution presented and explore a more general approach to solving such problems.

Problem Overview

The problem statement is: What is the larger number if the difference between the squares of two consecutive integers is 17?

Mathematical Approach to the Problem

Let's denote the smaller integer by ( x ) and the larger integer by ( x 1 ). The difference between their squares can be expressed as:

( (x 1)^2 - x^2 17 )

Expanding and simplifying the equation:

( (x^2 2x 1) - x^2 17 )

( 2x 1 17 )

( 2x 16 )

( x 8 )

Therefore, the two consecutive integers are ( 8 ) and ( 9 ). The larger of these numbers is ( 9 ).

Generalized Approach

Using a more generalized approach, if the difference between the squares of two consecutive integers ( n ) and ( n 1 ) is ( d ), then:

( (n 1)^2 - n^2 d )

( (n^2 2n 1) - n^2 d )

( 2n 1 d )

( 2n d - 1 )

( n frac{d - 1}{2} )

Therefore, ( n frac{d - 1}{2} ) and the larger integer ( (n 1) ) is:

( n 1 frac{d - 1}{2} 1 )

( frac{d 1}{2} )

For the specific case where ( d 17 ), substituting into the equation:

( n frac{17 - 1}{2} frac{16}{2} 8 )

( n 1 frac{17 1}{2} 9 )

Hence, the larger number is ( 9 ).

Verification

To verify, let's check the difference between the squares of ( 8 ) and ( 9 ):

( 9^2 - 8^2 81 - 64 17 )

The solution holds, confirming that ( 9 ) is indeed the larger number.

Using J Programming Language

For a more practical and programming-oriented approach, we can use the J programming language to solve the problem. Here's how you can do it:

Generate pairs of consecutive integers 0 to 20:

  2/i.20

Square the pairs:

  2^~a

Find the absolute value of the difference of each squared pair:

-/

Find and mark any pair of squared integers whose difference is 17:

Use the mark vector to find and list the pairs of original un-squared integers that when squared made a difference of 17:

a~

Find the larger integer of the pair:

  ./a~17-/

The answer is 9.

The J code can be run in any J environment to verify the solution:

  a : 2/i.20
  2^~a
  -/
  17
  a~17-/
  ./a~17-/

By executing the code, the output confirms that the larger integer is ( 9 ).

In conclusion, whether through mathematical reasoning or a programming approach, we can confidently determine that the larger integer whose squares have a difference of 17 is ( 9 ).