rajesh a money lender lends in the following manner. for the first year he charges 2% p.a, for the second year he charges 4% p.a, for the third year he charges 6% p.a and so on. if he lends a sum in this way at simple interest find the least integral number of years in which it will fetch an interestat least equal to itself

10 years

Since this is simple interest we dont calculate interest after adding interest.
Let the initial amount be 100
Thus in the first year the amount is 102 in the second year since the rate is 4%, This is 4% of 100 and not 102 Thus we add 4 to 102

Thus the amounts go like this
100 -> 102 -> 106 -> 112 -> 120 -> 130 -> 142 -> 156 -> 172 -> 190 -> 210

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!

To find the least integral number of years in which the interest will be at least equal to the sum lent, we need to set up an equation based on the given interest rates.

Let's assume the sum lent is 'x'.

After the first year, Rajesh charges 2% interest, which is equal to (2/100) * x = 0.02x.
After the second year, he charges 4% interest, which is equal to (4/100) * x = 0.04x.

In general, after the nth year, he charges (2n/100) * x = (0.02n) * x.

Now, we can set up the equation:

0.02x + 0.04x + 0.06x + ... + (0.02n) * x ≥ x

Simplifying the equation:

x * (0.02 + 0.04 + 0.06 + ... + 0.02n) ≥ x

Multiplying both sides by 100 to eliminate decimals:

2 + 4 + 6 + ... + 2n ≥ 100

Now, we need to find the least value of 'n' for which the sum of the series is greater than or equal to 100. This can be calculated using the formula for the sum of an arithmetic series:

(n/2) * (first term + last term) ≥ 100

Using the formula for the nth term of the series (2n), we can rewrite the equation:

(n/2) * (2 + 2n) ≥ 100

Simplifying further:

n + n^2 ≥ 50

Rearranging the equation:

n^2 + n - 50 ≥ 0

Factoring the equation:

(n + 10)(n - 5) ≥ 0

From this, we can conclude that 'n' must be greater than or equal to 5.

Therefore, the least integral number of years in which the interest will be at least equal to the sum lent is 5 years.

To find the least integral number of years in which the interest is at least equal to the sum lent, we can set up an equation and solve for the number of years.

Let's assume the sum lent is "P" (principal).

According to the given lending pattern, the interest rate for each year is increasing by 2%. Therefore, for the first year, the interest rate is 2%, for the second year it's 4%, and so on. So, the interest rate for the nth year is:

Interest Rate = 2n%

The interest earned for the nth year can be calculated as:

Interest = (Principal * Interest Rate) / 100

Since we want the interest earned to be at least equal to the principal, we can set up the following equation:

Principal <= (Principal * Interest Rate) / 100

Simplifying the equation, we get:

Principal <= (Principal * n * 2) / 100

Now, let's solve this equation to find the value of "n":

Principal <= (Principal * n * 2) / 100

Multiplying both sides by 100 to eliminate the denominator:

100 * Principal <= Principal * n * 2

Cancelling out the "Principal" term:

100 <= n * 2

Dividing both sides by 2:

50 <= n

This means that the least integral number of years in which the interest will be at least equal to the sum lent (Principal) is 50 years or more.

Therefore, the least integral number of years is 50 years.