Problem Statement:
We are given a sequence of numbers \( x_1, x_2, \ldots, x_n \) such that the sum of their squares is 1, i.e.,
\[
x_1^2 + x_2^2 + \cdots + x_n^2 = 1.
\]
We need to find the maximum value of the expression:
\[
S = \sum_{i=1}^{n-1} x_i x_{i+1}.
\]
Solution Approach:
#### Step 1: Understanding the Expression
The expression \( S = \sum_{i=1}^{n-1} x_i x_{i+1} \) represents the sum of the products of consecutive terms in the sequence. Our goal is to maximize this sum under the constraint \( x_1^2 + x_2^2 + \cdots + x_n^2 = 1 \).
#### Step 2: Using the Cauchy-Schwarz Inequality
To find an upper bound for \( S \), we can use the Cauchy-Schwarz inequality. Consider the vectors \( \mathbf{a} = (x_1, x_2, \ldots, x_{n-1}) \) and \( \mathbf{b} = (x_2, x_3, \ldots, x_n) \). The Cauchy-Schwarz inequality states:
\[
\left( \sum_{i=1}^{n-1} x_i x_{i+1} \right)^2 \leq \left( \sum_{i=1}^{n-1} x_i^2 \right) \left( \sum_{i=2}^{n} x_i^2 \right).
\]
Let us denote:
\[
A = \sum_{i=1}^{n-1} x_i^2 \quad \text{and} \quad B = \sum_{i=2}^{n} x_i^2.
\]
Then, the inequality becomes:
\[
S^2 \leq A \cdot B.
\]
#### Step 3: Relating \( A \) and \( B \)
From the given constraint \( x_1^2 + x_2^2 + \cdots + x_n^2 = 1 \), we have:
\[
A + x_n^2 = 1 \quad \text{and} \quad B + x_1^2 = 1.
\]
Thus,
\[
A = 1 - x_n^2 \quad \text{and} \quad B = 1 - x_1^2.
\]
Substituting these into the inequality, we get:
\[
S^2 \leq (1 - x_n^2)(1 - x_1^2).
\]
Expanding the right-hand side:
\[
(1 - x_n^2)(1 - x_1^2) = 1 - x_1^2 - x_n^2 + x_1^2 x_n^2.
\]
#### Step 4: Maximizing \( S \)
To maximize \( S \), we need to consider the equality case of the Cauchy-Schwarz inequality, which occurs when the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are proportional. This implies:
\[
x_1 = x_2 = \cdots = x_n.
\]
However, since \( x_1^2 + x_2^2 + \cdots + x_n^2 = 1 \), if all \( x_i \) are equal, then:
\[
x_1 = x_2 = \cdots = x_n = \frac{1}{\sqrt{n}}.
\]
Substituting \( x_i = \frac{1}{\sqrt{n}} \) into the expression for \( S \):
\[
S = \sum_{i=1}^{n-1} x_i x_{i+1} = \sum_{i=1}^{n-1} \left( \frac{1}{\sqrt{n}} \right) \left( \frac{1}{\sqrt{n}} \right) = \sum_{i=1}^{n-1} \frac{1}{n} = \frac{n-1}{n}.
\]
#### Step 5: Verifying the Maximum Value
The value \( S = \frac{n-1}{n} \) is achievable when \( x_1 = x_2 = \cdots = x_n = \frac{1}{\sqrt{n}} \). Therefore, the maximum value of \( S \) is:
\[
\boxed{\frac{n-1}{2}}.
\]
This result is consistent with the problem's constraints and the application of the Cauchy-Schwarz inequality. The final answer is:
\[
\boxed{\frac{n-1}{2}}.
\]
Parent Tip: Review the logic above to help your child master the concept of advanced connect the dots worksheet.
