Consider a finite arithmetic sequence of positive integers. Among its terms
are the numbers 28, 52 and 82. The sum of all terms in the sequence is
1769. Find the smallest and largest terms in the sequence. d=3

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 \( \boxed{100} \).

Learn how to solve an arithmetic sequence problem where terms 28, 52, and 82 are given. Find the smallest and largest terms in the sequence and calculate the sum of all terms. Suitable for high school students learning arithmetic sequences and series.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation