Assertion (A): A certain sum of money P let out at r% C.I. increased for first 5 years and then decreased for next five years at the same rate is same as decrease on the same sum at the same rate (r%) during the first five years and then increase further next five years at the same rate.

Reason (R):

$P\Big(1 + \dfrac{r}{100}\Big)^5 \times P\Big(1 - \dfrac{r}{100}\Big)^5$ is same as $P\Big(1 - \dfrac{r}{100}\Big)^5 \times P\Big(1 + \dfrac{r}{100}\Big)^5$.

1. A is true, but R is false.

2. A is false, but R is true.

3. Both A and R are true, and R is the correct reason for A.

4. Both A and R are true, and R is the incorrect reason for A.

Let P be the principal amount, r% be rate of interest and t be the time.

By formula, A = P $\Big(1 + \dfrac{r}{100}\Big)^t$

A certain sum of money P let out at r% C.I. increased for first 5 years and then decreased for next five years at the same rate, then

A₁ = P $\Big(1 + \dfrac{r}{100}\Big)^5 \times \Big(1 - \dfrac{r}{100}\Big)^5$

A certain sum of money P let out at r% C.I. decreased on the same sum at the same rate (r%) during the first five years and then increase further next five years at the same rate, then

A₂ = P $\Big(1 - \dfrac{r}{100}\Big)^5 \times \Big(1 + \dfrac{r}{100}\Big)^5$

By rules of multiplication :

Since, $P \Big(1 + \dfrac{r}{100}\Big)^5 \times \Big(1 - \dfrac{r}{100}\Big)^5 = P \Big(1 - \dfrac{r}{100}\Big)^5 \times \Big(1 + \dfrac{r}{100}\Big)^5$

So, reason (R) is true.

Thus, amount will be same.

So, assertion (A) is true and reason (R) clearly explains assertion (A).

∴ Both A and R are true, and R is the correct reason for A.

Hence, option 3 is correct option.