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:

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.

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.