1.     In an AP **16th Term** is three time the **5th term** and the **12th term** is **100.** What is the sum of the first twenty terms?

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!

In an AP 16th Term is three times the 5th term and the 12th term is 100. What is the sum of the first twenty terms?

This problem explores how to find the sum of the first 20 terms in an arithmetic progression where the 16th term is 3 times the 5th term and the 12th term is 100. Learn how to apply arithmetic progression formulas and algebraic techniques to solve this AP problem, suitable for students in Grades 9-11.

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