Let's solve the problem step-by-step.

Problem Recap:

We are tasked to determine the sum of the first 50 terms of an arithmetic sequence where:

• The 17th term $a_{17} = 53$,
• The 28th term $a_{28} = 86$,
• The sequence has a total of 50 terms.

Step 1: General formula for the $n$-th term of an arithmetic sequence

The general formula for the $n$-th term is: $a_n = a_1 + (n-1)d$ where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Step 2: Using the given terms to find the common difference $d$

From the given information:

1. $a_{17} = 53 \quad \Rightarrow \quad a_1 + 16d = 53$,
2. $a_{28} = 86 \quad \Rightarrow \quad a_1 + 27d = 86$.

Subtract the first equation from the second: $(a_1 + 27d) - (a_1 + 16d) = 86 - 53$ $11d = 33 \quad \Rightarrow \quad d = 3$

Step 3: Finding $a_1$ (the first term)

Substitute $d = 3$ into the first equation: $a_1 + 16d = 53$ $a_1 + 16(3) = 53 \quad \Rightarrow \quad a_1 + 48 = 53$ $a_1 = 5$

Step 4: Sum of the first 50 terms

The sum of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right)$ For $n = 50$, $a_1 = 5$, and $d = 3$: $S_{50} = \frac{50}{2} \left( 2(5) + (50-1)(3) \right)$ $S_{50} = 25 \left( 10 + 49(3) \right)$ $S_{50} = 25 \left( 10 + 147 \right)$ $S_{50} = 25 \times 157 = 3925$

Final Answer:

The sum of the first 50 terms of the arithmetic sequence is: $\boxed{3925}$

Would you like further clarification or additional examples?

5 Related Questions:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. What happens if we only know two terms of a sequence but their positions are unspecified?
3. How do we calculate the 50th term of the sequence explicitly?
4. Can this method apply to geometric sequences? If not, what would be different?
5. How does the value of $d$ affect the growth of the sum in an arithmetic sequence?

Tip:

When working with arithmetic sequences, always use the positions of known terms strategically to simplify calculations for $d$ and $a_1$.