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The image contains a set of problems on **Sequence and Series** and **Simple Interest and Compound Interest**. I will provide step-by-step solutions for a few of the questions.

Let's start with the **first question from Sequence and Series**:

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**1. Which term of the series in 6, 312½, … is –19?**

This appears to be an arithmetic progression (AP). The nth term of an AP can be given by the formula:

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

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

### Step 1: Identify the first term and common difference.
- The first term \( a = 6 \).
- The second term is \( 312 \frac{1}{2} \), or 312.5.
  
To find the common difference \( d \):
\[
d = 312.5 - 6 = 306.5
\]

### Step 2: Use the nth term formula to find \( n \) when \( a_n = -19 \).

We are given \( a_n = -19 \). Plugging in the values:

\[
-19 = 6 + (n - 1) \cdot 306.5
\]

Simplify the equation:

\[
-19 - 6 = (n - 1) \cdot 306.5
\]

\[
-25 = (n - 1) \cdot 306.5
\]

\[
n - 1 = \frac{-25}{306.5}
\]

\[
n - 1 \approx -0.0816
\]

\[
n \approx 0.9184
\]

Since \( n \) is approximately 1 (we expect an integer value), this suggests the term is very close to the first one but technically a fractional or zero term.

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Would you like to solve another question or go into more detail about this one? Here are five related questions:

1. How can we verify the common difference in arithmetic progressions?
2. What are the conditions for an arithmetic progression to have a negative term?
3. Can you explain how to find the sum of an arithmetic series?
4. What happens when the common difference is a fraction?
5. How would the solution change if we worked with a geometric series?

**Tip**: Always check if the series is arithmetic (constant difference) or geometric (constant ratio) before applying formulas.

The image contains math problems on Sequence and Series and Simple Interest and Compound Interest.

Explore detailed solutions to problems on Sequences and Series including AP and GP, along with Simple and Compound Interest calculations. Perfect for students in Grades 10-12.

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