The image contains a mathematical question asking for the statement and proof of the following:

1. Simple Interest Theorem
2. Compound Interest Theorem (as an alternative choice).

Here are detailed solutions:

1. Simple Interest Theorem

Statement:
Simple Interest is a financial calculation where the interest is calculated on the principal amount at a constant rate over time. The formula is:
$\text{SI} = \frac{P \times R \times T}{100}$ where:

• $P$ = Principal amount
• $R$ = Rate of interest per annum
• $T$ = Time in years

Proof:

1. Definition: Interest is the extra money paid for borrowing a sum (principal $P$) over time $T$ at a specified rate of interest $R$ per annum.

2. Derivation:

   • Interest for 1 year = $\frac{P \times R}{100}$
   • For $T$ years, multiply by $T$: $\text{SI} = \frac{P \times R \times T}{100}$

2. Compound Interest Theorem

Statement:
Compound Interest is calculated on the principal amount and also on the interest accumulated in previous periods. The formula is:
$A = P \left(1 + \frac{R}{100}\right)^T$ Where:

• $A$ = Total amount after $T$ years (Principal + Interest)
• $P$ = Principal amount
• $R$ = Rate of interest per annum
• $T$ = Time in years

The Compound Interest (CI) can be derived as: $\text{CI} = A - P = P \left(1 + \frac{R}{100}\right)^T - P$

Proof:

1. Definition: In Compound Interest, the interest for each year is added to the principal for the calculation of interest in the next year.

2. Derivation:

   • For 1 year: $A = P \left(1 + \frac{R}{100}\right)$
   • For 2 years: $A = P \left(1 + \frac{R}{100}\right) \cdot \left(1 + \frac{R}{100}\right) = P \left(1 + \frac{R}{100}\right)^2$
   • Generalizing for $T$ years: $A = P \left(1 + \frac{R}{100}\right)^T$
   • Subtracting $P$ from $A$ gives: $\text{CI} = A - P = P \left(1 + \frac{R}{100}\right)^T - P$

Would you like further details or examples?

Follow-up Questions:

1. How does compound interest differ from simple interest in real-life applications?
2. Can you calculate the simple and compound interest for $P = 10,000$, $R = 5\%$, $T = 3$ years?
3. What happens to compound interest if it is compounded quarterly or monthly instead of annually?
4. Explain the significance of the exponential growth in compound interest.
5. How does the duration of time (T) affect the difference between simple and compound interest?

Tip: When solving problems, always clarify the compounding period for compound interest calculations (annual, semi-annual, etc.).