A sum of money is paid back in two annual instalments of $\text{₹}1,764$ each, allowing $5\%$ compound interest compounded annually. The sum borrowed was:

Let the total sum borrowed be $P$.
The rate of interest is $R=5\%$ per annum.
The value of each annual instalment is $I=\text{₹}1,764$.

The formula for the principal amount in a compound interest two-instalment scheme is given by the sum of the present values of each instalment:

$$P=\frac{I}{{\left(1+\frac{R}{100}\right)}^1}+\frac{I}{{\left(1+\frac{R}{100}\right)}^2}$$

First, let's compute the fraction term $\left(1+\frac{5}{100}\right)$:

$$1+\frac{5}{100}=1+\frac{1}{20}=\frac{21}{20}$$

Now substitute this back into the formula:

$$P=\frac{1764}{\left(\frac{21}{20}\right)}+\frac{1764}{{\left(\frac{21}{20}\right)}^2}$$

$$P=1764\times \frac{20}{21}+1764\times \frac{400}{441}$$

Let's simplify each term:

* For the first term: $\frac{1764}{21}=84$

$$84\times 20=1680$$

* For the second term: $\frac{1764}{441}=4$

$$4\times 400=1600$$

Adding the two present values together to find the total sum borrowed:

$$P=1680+1600=\text{₹}3,280$$