To solve this problem, let's consider the given finite arithmetic sequence. The sequence has a common difference $d = 3$, and it contains the terms 28, 52, and 82. We are also given that the sum of all terms in the sequence is 1769.

Step 1: General Form of the Arithmetic Sequence

The general term of the arithmetic sequence can be written as: $a_n = a_1 + (n-1) \cdot d$ where $a_1$ is the first term, $d = 3$ is the common difference, and $n$ is the position of the term in the sequence.

Step 2: Express the Known Terms

We are given that 28, 52, and 82 are terms in the sequence. Let $a_i = 28$, $a_j = 52$, and $a_k = 82$ for some integers $i, j,$ and $k$. Then we can express these terms as: $28 = a_1 + (i-1) \cdot 3$ $52 = a_1 + (j-1) \cdot 3$ $82 = a_1 + (k-1) \cdot 3$

Step 3: Solve for the Differences Between Indices

Subtract the first equation from the second: [ 52 - 28 = [a_1 + (j-1) \cdot 3] - [a_1 + (i-1) \cdot 3] ] $24 = 3(j - i)$ $j - i = 8$

Now subtract the second equation from the third: [ 82 - 52 = [a_1 + (k-1) \cdot 3] - [a_1 + (j-1) \cdot 3] ] $30 = 3(k - j)$ $k - j = 10$

So, we have $j = i + 8$ and $k = j + 10 = i + 18$.

Step 4: Write the Terms Using $i$

Now, we can express the positions of these terms in terms of $i$: $a_i = 28, \quad a_{i+8} = 52, \quad a_{i+18} = 82$

Step 5: Sum of the Sequence

The sum of all terms in the sequence is given by: $S = \frac{n}{2} \cdot (a_1 + a_n)$ where $n$ is the total number of terms. We are given $S = 1769$.

Step 6: Calculate the First Term $a_1$

Since $a_i = 28$, we can express $a_1$ as: $28 = a_1 + (i-1) \cdot 3 \quad \text{so} \quad a_1 = 28 - 3(i-1)$ Substituting into the other terms: $52 = 28 + 3 \cdot 8 = 28 + 24 = 52 \quad \text{(this checks out)}$ $82 = 28 + 3 \cdot 18 = 28 + 54 = 82 \quad \text{(this checks out)}$

Step 7: Determine the Smallest and Largest Terms

To find the smallest and largest terms, note that the sequence starts at $a_1$ and ends at $a_n$. The largest term is $a_n = a_1 + (n-1) \cdot 3$.

Given $S = 1769$: $\frac{n}{2} \cdot (a_1 + a_n) = 1769$

Since we already have the three terms that fit in, the sequence should contain $n = 19$ terms (based on $i$ and $k$). The smallest term will be $a_1$ and the largest term $a_{19}$. We find:

$a_1 = 28 - 3(i-1) \quad \text{and} \quad a_{19} = a_1 + 54$

Thus, the sequence starts at 10 (smallest term) and ends at 82 (largest term).

Final Answer:

The smallest term is $\boxed{10}$ and the largest term is $\boxed{100}$.