The price of an article increases by 5% every year. If the difference between its price at the end of the second and the third year is ₹52.50, then what will be its price at the end of the first year?

Let the price of the article at the end of the first year be $$P$$ rupees.

Since the price rises by 5% every year, multiply by $$1 + \frac{5}{100} = 1.05$$ for each successive year.

Price at the end of the second year
$$P_2 = P \times 1.05$$

Price at the end of the third year
$$P_3 = P \times 1.05^2$$

The difference between these two prices is given to be ₹52.50:
$$P_3 - P_2 = 52.50$$

Substitute the expressions for $$P_3$$ and $$P_2$$:
$$P \times 1.05^2 - P \times 1.05 = 52.50$$

Factor out the common term $$P \times 1.05$$:
$$P \times 1.05 \left(1.05 - 1\right) = 52.50$$
$$P \times 1.05 \times 0.05 = 52.50$$

Simplify $$1.05 \times 0.05 = 0.0525$$, so
$$P \times 0.0525 = 52.50$$

Solve for $$P$$:
$$P = \frac{52.50}{0.0525} = 1000$$

Therefore, the price of the article at the end of the first year is ₹1,000.

Option A which is: ₹1,000