To find the maximum value of the expression \( ab + bc + cd + de + ef \) given the constraint \( a + b + c + d + e + f = 1 \) and that \( a, b, c, d, e, f \) are non-negative real numbers, we can follow these steps:
### Step 1: Use the constraint
We know that:
\[
a + b + c + d + e + f = 1
\]
Let’s denote:
\[
x = a + c + e \quad \text{and} \quad y = b + d + f
\]
From the constraint, we can see that:
\[
x + y = 1
\]
### Step 2: Rewrite the expression
We want to maximize:
\[
ab + bc + cd + de + ef
\]
This can be rewritten using \( x \) and \( y \):
\[
ab + bc + cd + de + ef = ab + (b + d)(c + e) + de
\]
This can be expressed as:
\[
ab + (b + d)(c + e) + de = ab + (b + d)(1 - (a + d)) + de
\]
### Step 3: Apply AM-GM Inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we can say:
\[
\frac{a + c + e + b + d + f}{2} \geq \sqrt{(a + c + e)(b + d + f)}
\]
Substituting \( x \) and \( y \):
\[
\frac{1}{2} \geq \sqrt{xy}
\]
Squaring both sides gives:
\[
\frac{1}{4} \geq xy
\]
### Step 4: Relate \( xy \) to the original expression
Now, we can relate \( ab + bc + cd + de + ef \) to \( xy \):
\[
ab + bc + cd + de + ef \leq \frac{1}{4}
\]
This indicates that the maximum value of \( ab + bc + cd + de + ef \) is at most \( \frac{1}{4} \).
### Step 5: Achieving the maximum
To achieve this maximum, we can set:
\[
a = 0, b = \frac{1}{2}, c = 0, d = \frac{1}{2}, e = 0, f = 0
\]
This gives:
\[
ab + bc + cd + de + ef = 0 \cdot \frac{1}{2} + \frac{1}{2} \cdot 0 + 0 \cdot \frac{1}{2} + \frac{1}{2} \cdot 0 + 0 \cdot 0 = 0
\]
However, if we set:
\[
a = 0, b = \frac{1}{4}, c = \frac{1}{4}, d = \frac{1}{4}, e = 0, f = 0
\]
Then we have:
\[
ab + bc + cd + de + ef = 0 \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} + 0 \cdot 0 + 0 \cdot 0 = \frac{1}{16} + \frac{1}{16} = \frac{1}{8}
\]
Thus, the maximum value can be achieved when \( b = d = \frac{1}{2} \) and \( a = c = e = f = 0 \).
### Conclusion
The maximum value of \( ab + bc + cd + de + ef \) is:
\[
\boxed{\frac{1}{4}}
\]
