A certain sum is deposited for 4 years at a rate of 10% per annum on compound interest compounded annually. The difference between the interest at the end of 2 years and that at the end of 4 years is ₹5,082. Find the sum (in ₹).

The correct option is 20,000.

Let us find the sum step-by-step using the formula for compound interest.

Let the principal sum be denoted by P (in ₹).
The rate of interest is r = 10% per annum, compounded annually.
The formula for the amount A_t accumulated after t years under annual compound interest is given by:

$A_t=P{(1+\frac{r}{100})}^t$

Substituting r = 10 into the formula, we get:

$A_t=P{(1+\frac{10}{100})}^t=P{\left(1.1\right)}^t$

The compound interest CI_t at the end of t years is the accumulated amount minus the principal:

$C{I}_t=A_t-P=P[{\left(1.1\right)}^t-1]$

Let us calculate the compound interest at the end of 2 years (t = 2):

$C{I}_2=P[{\left(1.1\right)}^2-1]=P[1.21-1]=0.21P$

Now, let us calculate the compound interest at the end of 4 years (t = 4):

$C{I}_4=P[{\left(1.1\right)}^4-1]=P[1.4641-1]=0.4641P$

According to the problem, the difference between the compound interest at the end of 4 years and 2 years is ₹5,082:

$C{I}_4-C{I}_2=\mathrm{5,082}$

Substitute the expressions in terms of P:

$0.4641P-0.21P=\mathrm{5,082}$

$0.2541P=\mathrm{5,082}$

Now, we solve for P:

$P=\frac{\mathrm{5,082}}{0.2541}$

Multiplying both numerator and denominator by 10,000 to clear the decimal:

$P=\frac{\mathrm{50,820,000}}{\mathrm{2,541}}$

Since 2,541 * 2 = 5,082, we get:

$P=\mathrm{20,000}$

Thus, the principal sum is ₹20,000.