Let's go through the given problem step-by-step.

Problem:

We are given an arithmetic sequence that includes the numbers 28, 52, and 82. The sum of all terms in this sequence is 1769. We need to find the smallest and largest terms in the sequence.

Key points:

1. The sequence is arithmetic (the difference between consecutive terms is constant).
2. The sum of the terms is 1769.
3. Some known terms in the sequence are 28, 52, and 82.

General form of an arithmetic sequence:

Let:

• $a$ be the first term,
• $d$ be the common difference,
• $n$ be the number of terms.

The general term of an arithmetic sequence is given by: $T_k = a + (k-1) \cdot d$ where $T_k$ is the $k$-th term.

Given dataLet's go through the given problem step-by-step.

Problem:

We are given an arithmetic sequence that includes the numbers 28, 52, and 82. The sum of all terms in this sequence is 1769. We need to find the smallest and largest terms in the sequence.

Key points:

1. The sequence is arithmetic (the difference between consecutive terms is constant).
2. The sum of the terms is 1769.
3. Some known terms in the sequence are 28, 52, and 82.

General form of an arithmetic sequence:

Let:

• $a$ be the first term,
• $d$ be the common difference,
• $n$ be the number of terms.

The general term of an arithmetic sequence is given by: $T_k = a + (k-1) \cdot d$ where $T_k$ is the $k$-th term.

Given

• $T_i = 28$
• $T_j = 52$
• $T_k = 82$
• The sum of all terms is 1769.

The sum of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a + l)$ where $S_n$ is the sum, $l$ is the last term, and $n$ is the total number of terms.

Let's now find the relation between $a$ and $d$ using the given terms 28, 52, and 82. Then, we can determine the number of terms and compute the smallest and largest terms.

I'll calculate that now.It looks like I forgot to define the positions of the given terms. Let's assume:

• The position of 28 is $T_i$,
• The position of 52 is $T_j$,
• The position of 82 is $T_k$.

Since 28, 52, and 82 are distinct numbers in the sequence, let's assign their positions as $i = 1$, $j = 2$, and $k = 3$ for simplicity. Now I'll solve this system of equations.It seems like I can't do more advanced data analysis right now. Please try again later.

In the meantime, if you want, I can guide you through the solution manually or explain any concepts related to arithmetic sequences that might help. Would you like me to proceed in that way?

Here are some related questions for further exploration:

1. What is the formula for finding the sum of an arithmetic sequence?
2. How do we find the common difference $d$ in an arithmetic sequence?
3. Can an arithmetic sequence have both positive and negative common differences?
4. How would the solution change if we knew the number of terms in the sequence?
5. What are some other real-life applications of arithmetic sequences?

Tip: Always keep in mind that for arithmetic sequences, knowing at least two terms allows you to calculate the common difference.