Assertion (A): The amount on ₹ 4000 at 10% p.a compounded annually for $2\dfrac{1}{2}$ years = $4000\Big(1+\dfrac{10}{100}\Big)^\dfrac{5}{2}$.

Reason (R): When the total time is not complete number of years, for example time = m years and n months, and rate is r % p.a (compounded annually), then

A = sum $\Big(1+\dfrac{r}{100}\Big)^m \times \Big(1+\dfrac{\dfrac{r}{12}}{100}\Big)^n$

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are false.

Explanation

Given,

P = ₹ 4,000, R = 10%, T = 2 year

A = P x $\Big(1 + \dfrac{R}{100}\Big)^T$

$\text{Amount in 2 years} = 4,000 \times \Big(1 + \dfrac{10}{100}\Big)^{2}\\[1em] = 4000 \times \Big(1 + \dfrac{1}{10}\Big)^{2}\\[1em] = 4000 \times \Big(\dfrac{11}{10}\Big)^2\\[1em] = 4000 \times \Big(\dfrac{121}{100}\Big)\\[1em] = ₹ 4840$

After two years, ₹ 4840 is the principal for the remaining half year and so:

$\text{A} = P \times \Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2}\\[1em] = 4840 \times \Big(1 + \dfrac{10}{2 \times 100}\Big)^{\dfrac{1}{2} \times 2}\\[1em] = 4840 \times \Big(\dfrac{20 + 1}{20}\Big)^{\dfrac{1}{2} \times 2}\\[1em] = 4840 \times \Big(\dfrac{21}{20}\Big)\\[1em] = ₹ 5082$

According to Assertion,

$= 4,000\Big(1 + \dfrac{10}{100}\Big)^\dfrac{5}{2}\\[1em] = 4,000\Big(1 + \dfrac{1}{10}\Big)^\dfrac{5}{2}\\[1em] = 4,000\Big(1 + 0.1\Big)^\dfrac{5}{2}\\[1em] = 4,000 \times (1.1)^\dfrac{5}{2}\\[1em] = 4,000 \times (\sqrt{1.1})^5\\[1em] = 4,000 \times (1.05)^5\\[1em] = 4,000 \times 1.28\\[1em] = 5,105.12$

Hence, Amount in $2\dfrac{1}{2}$ years = ₹ 5082 ≠ 5,105.12

∴ Assertion (A) is false.

A = sum $\Big(1+\dfrac{r}{100}\Big)^m \times \Big(1+\dfrac{\dfrac{r}{12}}{100}\Big)^n$

$= 4,000 \Big(1+\dfrac{10}{100}\Big)^2 \times \Big(1+\dfrac{\dfrac{10}{12}}{100}\Big)^6\\[1em] = 4,000 \Big(1+\dfrac{1}{10}\Big)^2 \times \Big(1+\dfrac{10}{1200}\Big)^6\\[1em] = 4,000 \times (1+0.1)^2 \times (1+0.0083)^6\\[1em] = 4,000 \times(1.1)^2 \times 1.051\\[1em] = 4,000 \times 1.21 \times 1.051\\[1em] = 5076$

∴ Reason (R) is false.

∴ Hence, both Assertion (A) and Reason (R) are false.