To solve this problem using the concept of geometric progression and its summation formula, let's break it down step-by-step.

Formula for Compound Interest:

The total amount $A$ after $n$ years with compound interest is given by the formula:

$A = P(1 + r)^n$ Where:

• $P$ is the principal amount (initial investment),
• $r$ is the annual interest rate (expressed as a decimal),
• $n$ is the number of years,
• $A$ is the total amount after $n$ years.

Given Values:

• $P = 200,000$,
• $r = 5\% = 0.05$,
• $n = 6$.

a) Find the total amount of money after 6 years:

We need to apply the formula for compound interest:

$A = 200,000 \times (1 + 0.05)^6$

Now, let’s calculate the value:

$A = 200,000 \times (1.05)^6$ $A = 200,000 \times 1.3401$ $A = 268,020$

So, the total amount after 6 years would be Rs. 268,020.00.

b) Explanation Using Geometric Progression:

The technique I used involves recognizing that compound interest forms a geometric progression. In each year, the amount of money grows by a fixed ratio, $1 + r$, which is the characteristic of a geometric progression.

The sum of the amount at each year can be seen as:

$S_n = P \times (1 + r)^n$

This is a sum of terms where the common ratio is $1 + r$ and each successive term represents the amount accumulated at the end of each year. In this case, we only need the formula for the last term, $A$, as we are looking for the total value after 6 years.

Thus, this formula directly gives us the total accumulated amount without needing to calculate each year's interest separately. The result comes from the geometric progression of each year's compounded value.

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Would you like more details on any step or concepts?

Here are some related questions for further exploration:

1. How would the total amount change if the interest rate were 7% instead of 5%?
2. How do you derive the formula for compound interest using the geometric progression method?
3. What is the difference between simple interest and compound interest in terms of geometric progression?
4. How would you solve this problem using logarithms for a quicker computation?
5. How does the principal amount impact the total interest earned in compound interest scenarios?

Tip: To calculate compound interest efficiently, using a financial calculator or a spreadsheet can significantly speed up the process.