In an AP the 16th Term is three time the 5th Term and the 12th Term is 100  
 
	What is the sIn an AP the 16th Term is three time the 5th Term and the 12th Term is 100. What is the sum of the first twenty terms?  	 
 
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We are given the following information about an Arithmetic Progression (AP):

1. The 16th term is three times the 5th term.
2. The 12th term is 100.
3. We need to find the sum of the first 20 terms of this AP.

### Step 1: Formula for the nth term of an AP

The general formula for the nth term of an AP is:
\[
T_n = a + (n - 1) d
\]
where:
- \( T_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 2: Expressing the 16th term and 5th term

Using the formula for the nth term:
- The 16th term \( T_{16} = a + 15d \),
- The 5th term \( T_5 = a + 4d \).

From the problem statement, we know that the 16th term is three times the 5th term:
\[
a + 15d = 3(a + 4d)
\]
Simplifying this equation:
\[
a + 15d = 3a + 12d
\]
\[
15d - 12d = 3a - a
\]
\[
3d = 2a
\]
\[
a = \frac{3d}{2}
\]  \(\quad \text{(Equation 1)}\)

### Step 3: Using the information about the 12th term

The 12th term is given as 100:
\[
T_{12} = a + 11d = 100
\]
Substitute \( a = \frac{3d}{2} \) into this equation:
\[
\frac{3d}{2} + 11d = 100
\]
Multiply through by 2 to eliminate the fraction:
\[
3d + 22d = 200
\]
\[
25d = 200
\]
\[
d = 8
\]

### Step 4: Finding the first term \( a \)

Substitute \( d = 8 \) into \( a = \frac{3d}{2} \):
\[
a = \frac{3 \times 8}{2} = 12
\]

### Step 5: Finding the sum of the first 20 terms

The sum of the first \( n \) terms of an AP is given by:
\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\]
Substitute \( n = 20 \), \( a = 12 \), and \( d = 8 \):
\[
S_{20} = \frac{20}{2} \times (2 \times 12 + 19 \times 8)
\]
\[
S_{20} = 10 \times (24 + 152)
\]
\[
S_{20} = 10 \times 176
\]
\[
S_{20} = 1760
\]

### Final Answer:
The sum of the first 20 terms of the AP is **1760**.

Would you like further clarification or details on any step?

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Here are 5 related questions to expand on this:

1. How would the formula change if we needed to find the sum of the first 30 terms?
2. What is the common difference in an arithmetic progression if the sum of the first 10 terms is 145?
3. How do you find the nth term in a geometric progression?
4. How can you determine the common difference in an AP if only the first and last terms are known?
5. What is the sum of the first 15 terms of an AP where the 10th term is 50 and the common difference is 5?

**Tip**: When solving AP problems, always start by writing the nth term formula and substitute the given values step by step.

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

Learn how to find the sum of the first 20 terms in an arithmetic progression where the 16th term is three times the 5th term, and the 12th term is 100. This detailed solution includes the formula for the nth term and sum of an AP. Suitable for Grades 9-10.

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