Finding the Greatest Consecutive Odd Integers Whose Sum is at Most 20

In this article, we will explore how to determine the greatest possible consecutive odd integers whose sum does not exceed 20. This problem involves understanding the properties of odd integers and using mathematical reasoning to find the solution. Let's break down the problem step by step.

Understanding the Problem

The problem states that the sum of two consecutive odd integers is at most 20. We need to identify the greatest possible values of these integers.

Solving the Problem

Let's denote the first odd integer as ( n ). The next consecutive odd integer would be ( n 2 ). According to the problem, the sum of these two integers should be at most 20. We can set up the following inequality:

[[n (n 2) leq 20]

Simplifying the inequality:

[[2n 2 leq 20] [[2n leq 18] [[n leq 9]

Therefore, the first odd integer ( n ) can be at most 9. Consequently, the next consecutive odd integer would be ( n 2 9 2 11 ).

Verification

Let's verify the solution by calculating the sum of these two integers:

[[9 11 20]

This confirms that the sum is exactly 20, which is the maximum possible sum under the given conditions.

Alternative Approach: Using Nearest Odd Integers

Another approach to solving this problem is to consider the nearest odd integers around half of 20. Dividing 20 by 2 gives us 10. The nearest odd integers to 10 are 9 and 11. Let's verify the sum:

[[9 11 20]

Again, the sum is exactly 20, confirming that 9 and 11 are the greatest possible consecutive odd integers whose sum is at most 20.

Formal Solution Using Variables

We can also use variables to represent the consecutive odd integers. Let ( k ) be an integer. The two consecutive odd integers can be written as ( 2k - 1 ) and ( 2k 1 ). The sum of these integers is:

[[(2k - 1) (2k 1) 4k]

The problem states that the sum should be at most 20:

[[4k leq 20] [[k leq 5]

Since ( k ) is an integer, the greatest possible value for ( k ) is 5. Therefore, the two consecutive odd integers are:

[[2(5) - 1 9] [[2(5) 1 11]

Thus, the greatest possible consecutive odd integers whose sum is at most 20 are 9 and 11.

Conclusion

In conclusion, the problem of finding the greatest possible consecutive odd integers whose sum is at most 20 can be solved through various methods, including direct calculation, considering the nearest odd integers, and using formal mathematical reasoning with variables. The solution is 9 and 11.

Remember, understanding the properties of odd integers and applying basic algebraic principles can help you solve similar problems efficiently.