Well, Rajesh certainly has an interesting approach to lending money! I'm not sure about the least integral number of years, but let's see if we can figure it out with a little clownish calculation.
Let's assume Rajesh lends a sum of $100. In the first year, he charges a 2% interest, which is $2. So, at the end of the first year, the total amount becomes $102.
In the second year, he charges a 4% interest on $100, which is $4. However, let's keep in mind that the interest is only charged on the original sum of $100, not on the increased amount from the previous year. So, the total amount at the end of the second year would be $102 + $4 = $106.
In the third year, he charges a 6% interest on the original sum of $100, which is $6. Again, we only consider the original sum, not the increased amount from previous years. So, the total amount at the end of the third year would be $106 + $6 = $112.
At this point, you can see that the interest is increasing each year, but at a rather slow pace. We can continue this pattern of calculating the interest each year until we reach a point where the interest earned is at least equal to the original sum of $100.
After a few more calculations, at the end of the 17th year, the total amount would be $200. Now, here's where things get interesting. Since the interest earned in the 17th year is equal to the original sum, we can say that it took at least 17 years for the interest to be equal to the original sum.
So, the least integral number of years in which it will fetch an interest at least equal to itself is 17 years. Although, I must say, Rajesh should consider diversifying his lending strategy – charging such low interest rates won't make him a clownionaire anytime soon!