In this problem, we are given an arithmetic progression (AP) with the following information:

1. The 16th term is three times the 5th term.
2. The 12th term is 100.

We need to find the sum of the first twenty terms of this AP.

Step 1: Formula for the $n$-th term of an AP

The general formula for the $n$-th term $T_n$ of an arithmetic progression is: $T_n = a + (n - 1)d$ Where:

• $a$ is the first term.
• $d$ is the common difference.

Step 2: Using the given conditions

1. 16th term is three times the 5th term: $T_{16} = 3 \cdot T_5$ Using the formula for the $n$-th term, we can express $T_{16}$ and $T_5$: $a + 15d = 3 \cdot (a + 4d)$ Simplifying: $a + 15d = 3a + 12d$ Rearranging terms: $15d - 12d = 3a - a$ $3d = 2a$ $a = \frac{3d}{2}$

2. 12th term is 100: $T_{12} = a + 11d = 100$ Substitute $a = \frac{3d}{2}$ into the equation: $\frac{3d}{2} + 11d = 100$ Multiply through by 2 to eliminate the fraction: $3d + 22d = 200$ $25d = 200$ $d = 8$ Now, substitute $d = 8$ into $a = \frac{3d}{2}$: $a = \frac{3 \times 8}{2} = 12$

Step 3: Sum of the first 20 terms

The sum of the first $n$ terms of an arithmetic progression is given by: $S_n = \frac{n}{2} \cdot (2a + (n-1)d)$ For $n = 20$, $a = 12$, and $d = 8$: $S_{20} = \frac{20}{2} \cdot (2 \cdot 12 + (20-1) \cdot 8)$ $S_{20} = 10 \cdot (24 + 19 \cdot 8)$ $S_{20} = 10 \cdot (24 + 152) = 10 \cdot 176 = 1760$

Final Answer:

The sum of the first twenty terms is 1760.

---

Would you like further details on any part of the solution? Here are five related questions:

1. How do you find the common difference in an arithmetic progression?
2. What is the formula for the sum of an arithmetic series?
3. Can an arithmetic progression have a negative common difference?
4. How can you verify if a given sequence is an arithmetic progression?
5. What happens if the common difference in an arithmetic progression is zero?

Tip: Always verify each step with the given conditions to avoid errors in AP problems!