To solve these problems, we will apply two different formulas.

Part (a) Formula: Future Value of an Ordinary Annuity

For the first problem, you are investing ₹10,000 at the end of each year, which makes it an ordinary annuity.

The formula for the future value of an ordinary annuity is:

$FV = P \times \left[ \frac{(1 + i)^n - 1}{i} \right]$

Where:

• $P = 10,000$ (annual investment),
• $i = 0.05$ (interest rate per year),
• $n = 10$ (number of years).

Step 1: Substitute the values into the formula

$FV = 10,000 \times \left[ \frac{(1 + 0.05)^{10} - 1}{0.05} \right]$

$FV = 10,000 \times \left[ \frac{(1.628894626) - 1}{0.05} \right]$

$FV = 10,000 \times \left[ \frac{0.628894626}{0.05} \right]$

$FV = 10,000 \times 12.57789252 = 1,25,778.92$

So, the total amount you will have at the end of 10 years is ₹1,25,778.92.

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Part (b) Formula: Future Value with Lump Sum Deposits at the Beginning of the Year

For the second problem, you are investing ₹20,000 at the beginning of each year, which uses the future value formula for annuity due.

To account for this, we multiply the ordinary annuity formula by $(1 + i)$, because payments are made at the beginning of each period.

$FV = P \times \left[ \frac{(1 + i)^n - 1}{i} \right] \times (1 + i)$

Where:

• $P = 20,000$,
• $i = 0.04$,
• $n = 7$.

Step 2: Substitute the values into the formula

$FV = 20,000 \times \left[ \frac{(1 + 0.04)^7 - 1}{0.04} \right] \times (1 + 0.04)$

$FV = 20,000 \times \left[ \frac{(1.316074012) - 1}{0.04} \right] \times 1.04$

$FV = 20,000 \times \left[ \frac{0.316074012}{0.04} \right] \times 1.04$

$FV = 20,000 \times 7.901850299 \times 1.04 = 1,64,319.08$

So, the total amount you will have at the end of 7 years is ₹1,64,319.08.

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Do you want more details or have any questions?

Here are some related questions:

1. What is the difference between an ordinary annuity and an annuity due?
2. How would the final amount change if the interest rate was 6% for the first problem?
3. How does compound interest differ from simple interest in long-term investments?
4. If the investments were made monthly instead of annually, what formulas would we use?
5. What happens to the final amount if you invest for 20 years instead of 10 or 7 years?

Tip: In annuities, the timing of payments (beginning vs. end of period) significantly affects the future value due to compounding.