To solve this problem, let’s break down the information provided:

1. Given Data:

   • This is an arithmetic progression (AP).
   • The 15th term ($a_{15}$) is 18.
   • The 17th term ($a_{17}$) is 14.

2. AP Formula: For an arithmetic progression, the $n$-th term is given by: $a_n = a + (n - 1) \cdot d$ where $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

3. Set up equations: Using the term formula:

   • For the 15th term: $a + 14d = 18$
   • For the 17th term: $a + 16d = 14$

4. Solve for $d$: Subtract the first equation from the second to eliminate $a$: $(a + 16d) - (a + 14d) = 14 - 18$ $2d = -4 \Rightarrow d = -2$

5. Find $a$: Substitute $d = -2$ into one of the original equations, say $a + 14d = 18$: $a + 14(-2) = 18$ $a - 28 = 18 \Rightarrow a = 46$

6. Sum of the First 50 Terms: The sum of the first $n$ terms of an AP is given by: $S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d)$ Here, $n = 50$, $a = 46$, and $d = -2$.

   Substitute these values into the sum formula: $S_{50} = \frac{50}{2} \cdot (2 \cdot 46 + (50 - 1) \cdot (-2))$ $= 25 \cdot (92 - 98)$ $= 25 \cdot (-6)$ $= -150$

Final Answer:

The sum of the first 50 terms is -150.

Would you like a deeper explanation of any step or have any questions?

Here are some related questions to deepen understanding:

1. How would you find the 50th term of this AP?
2. Can you determine the sum of terms from the 10th to the 20th term?
3. How would the sum formula change if this were a geometric sequence instead of an arithmetic one?
4. If only the 15th term and the 50th term were given, how could you find the common difference and the first term?
5. How does the negative common difference affect the behavior of the sequence?

Tip: When dealing with arithmetic progressions, the difference between any two terms provides a direct way to calculate the common difference $d$ quickly.