91Ó°ÊÓ

Natural numbers are the backbone of this problem. They include all positive integers starting from 1, 2, 3, and so on.
This sequence goes on infinitely.
In our exercise, we deal with a natural number 'n', where 'n' is greater than or equal to 2.
Natural numbers do not include zero or any negative numbers. These numbers are very familiar as they are typically the kind we first learn to count with.
They are used to quantify and order objects in the real world.
For example:

• If n = 4, then there are four numbers in the sequence, such as 1, 2, 3, 4.
• The condition \( n \geq 2 \) guarantees that we are working with at least two elements.

Understanding these numbers helps us set the stage for dealing with more complex mathematical constructs.

Real numbers encompass a broader set that includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
They can be thought of as points on an infinite line, commonly called the number line.
In this exercise, the numbers \( x_1, x_2, ..., x_n \) are all real numbers.
They follow these properties:

• They are arranged in non-decreasing order, meaning each term is equal to or greater than the one before it: \( 0
• Their sum is exactly 1: \( \sum_{i=1}^n x_i = 1 \).
• The largest term is not greater than \( 2/3 \): \( x_n \leq 2/3 \).

Real numbers are very important in mathematics because they include both rational numbers (numbers that can be expressed as a fraction) and irrational numbers (numbers that cannot be expressed as fractions, like \( \pi \) and \( \sqrt{2} \)). This gives us a comprehensive way to discuss and measure quantities, magnitudes, and distances in various contexts.

Partial sums are sums of the first 'k' terms of a sequence.
They are very useful for understanding the intermediate stages of accumulation in a series.
For this problem, we define partial sums as follows: \( S_k = \sum_{j=1}^k x_j \).
This means:

• \( S_1 = x_1 \)
• \( S_2 = x_1 + x_2 \)
• \( S_3 = x_1 + x_2 + x_3 \)
• And so on, until \( S_n = x_1 + x_2 + ... + x_n \).

In our exercise, we deduced that the partial sums must include a value between 1/3 and 2/3.
This is a crucial point because it establishes the existence of a specific 'k' where \( 1/3 \leq S_k < 2/3 \).
To find such a 'k', we use the Intermediate Value Theorem, ensuring that the sequence passes through the required range.

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus.
It states that if a continuous function, f, takes on values f(a) and f(b) at two points a and b, then it also takes on any value between f(a) and f(b) at some point between a and b.
In our problem, let's apply the IVT to the partial sums:
Suppose we have a sequence \( S_k = \sum_{j=1}^k x_j \), where \( S_1, S_2, ..., S_n \) are the partial sums and \( S_n = 1 \).
According to the given conditions: \( S_n = 1 \) and \( S_k < 1/3 \) for all \( 1 \leq k < n \).
The IVT tells us that since \( S_n = 1 \leq 2/3 \) and \( S_1 \) started from a smaller value, there must be a value 'k' such that \( 1/3 \leq S_k < 2/3 \).
This is important because it helps us bridge the gap in our sequence, ensuring that every intermediate value is covered.
In this case, it leads us to the proof required by the exercise, affirming that such a 'k' indeed exists.