In an AP the 10th Term is twice the 4th Term and the 13th Term is 60.
What is the sum of the first twenty terms? 


Let's break this down step by step.

### Step 1: Use the formula for the \(n\)-th term of an arithmetic progression (AP).
The formula for the \(n\)-th term of an AP is:

\[
T_n = a + (n-1) \cdot d
\]

where:
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

### Step 2: Set up the equations based on the given conditions.

**1. 10th Term is twice the 4th Term:**

The 10th term is \(T_{10}\), and the 4th term is \(T_4\).

\[
T_{10} = a + (10-1) \cdot d = a + 9d
\]
\[
T_4 = a + (4-1) \cdot d = a + 3d
\]

The condition states that the 10th term is twice the 4th term, so:

\[
a + 9d = 2 \cdot (a + 3d)
\]

Simplifying this equation:

\[
a + 9d = 2a + 6d
\]
\[
9d - 6d = 2a - a
\]
\[
3d = a
\]

So, we have the relation:

\[
a = 3d \quad \text{(Equation 1)}
\]

**2. 13th Term is 60:**

The 13th term is \(T_{13}\), and we know \(T_{13} = 60\):

\[
T_{13} = a + (13-1) \cdot d = a + 12d = 60
\]

Substituting \(a = 3d\) from Equation 1:

\[
3d + 12d = 60
\]
\[
15d = 60
\]
\[
d = 4
\]

### Step 3: Find the value of \(a\).

Using \(a = 3d\):

\[
a = 3 \cdot 4 = 12
\]

### Step 4: Find the sum of the first 20 terms.

The sum of the first \(n\) terms of an AP is given by:

\[
S_n = \frac{n}{2} \cdot (2a + (n-1) \cdot d)
\]

For the first 20 terms:

\[
S_{20} = \frac{20}{2} \cdot (2a + (20-1) \cdot d)
\]
\[
S_{20} = 10 \cdot (2 \cdot 12 + 19 \cdot 4)
\]
\[
S_{20} = 10 \cdot (24 + 76)
\]
\[
S_{20} = 10 \cdot 100 = 1000
\]

### Final Answer:
The sum of the first 20 terms is **1000**.

---

Would you like a more detailed explanation or further questions on this? Here are some related questions to explore:

1. How do you derive the general formula for the sum of the first \(n\) terms of an AP?
2. How would the sum of the first 20 terms change if the common difference \(d\) were different?
3. What would happen to the 13th term if the first term were doubled?
4. How can you apply this method to solve problems involving the nth term and sum of other types of sequences?
5. How do you solve problems involving quadratic sequences?

**Tip**: The key to solving these kinds of problems is to use the properties of arithmetic progressions and express the terms in terms of \(a\) and \(d\), then manipulate the equations to solve for the unknowns.

In an AP the 10th Term is twice the 4th Term and the 13th Term is 60. What is the sum of the first twenty terms?

Learn how to find the sum of the first 20 terms in an arithmetic progression (AP) given that the 10th term is twice the 4th term and the 13th term is 60. This problem involves using the formula for the nth term of an AP and the sum formula. Suitable for students in Grades 8-10.

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