Hi Prathyush,

The sum of n terms in an AP = $$\frac{n}{2}\left[first\ term\ +\ last\ term\right]$$

$$Average=\frac{sum\ ofthe\ observations}{number\ of\ observations}$$

$$\therefore\ $$ The sum of n odd terms in an AP = $$\frac{n}{2}\left[3+2n+1\right]=\frac{n}{2}\left[2n+4\right]=n\left(n+2\right)$$

Average of n odd numbers = $$\frac{n\left(n+2\right)}{n}=n+2$$

The sum of n even terms in an AP = $$\frac{n}{2}\left[2+2n\right]=\ n\ \left(n+1\right)$$

Average of n even numbers = $$\frac{n\left(n+1\right)}{n}=n+1$$

Hi Rafa,
Yes, we will pass the referral amount of E to D, and D to C, and C to B, and B to A in the following manner-
Person A receives a 25% commission directly from the revenue, which amounts to $$5,000\ $$ for a total revenue of $20,000.

Person B, referred by A. A also receives 25% of B's commission, which equals 5000/4 = $$1,250\ $$.

Person C, referred by B. C's commission, is $$5,000\ $$, and B receives 25% of C's commission, which is another 5000/4. A, in turn, receives 25% of B's commission due to C, which is 5000/4*4
Person D, referred by C. D's commission, is $$5,000\ $$, and C receives 25% of D's commission, which is another 5000/4. B, in turn, receives 25% of C's commission due to D, which is 5000/4*4. A, in turn, receives 25% of B's commission due to C, which is 5000/4*4*4
This pattern continues, with each new person (E, F, and so on) contributing to the commissions of those who referred them.

To calculate A's total earnings, we can see that it follows a geometric progression (G.P.):

Total earnings for A = Sum to infinite G.P. with the initial term (a) of $$5,000\ $$ and a common ratio (r) of $$\frac{1}{4}\ $$.

Total earnings = $$\frac{a}{1-r} = \frac{5,000}{1-\frac{1}{4}} = \frac{20,000}{3} = Rs. 6,666.67\ $$.
I hope this helps!

Hi Sanskriti Agarwal,
The chocolates are being exchanged between themselves. So, the total
number of chocolates with them will always be the same. It is also mentioned
that the ratio is the same after every transaction. It means the number of
chocolates with either of them will be the same after every transaction.

Ex, There are 200 chocolates.

Let the ratio of chocolates with John and Abraham after every transaction be 2:3.

After 1st transaction, Chocolates with John = $$\frac{2}{5}\times\ 200\ =\ 80$$.

After nth transaction, Chocolates with John = $$\frac{2}{5}\times\ 200\ =\ 80$$.

As the ratio is the same and the total number of chocolates is constant,
both of them will have the same number of chocolates after every
transaction.
The total number of chocolates here is constant. The only way in which the ratio of chocolates with them always remains the same is if the keep giving an equal number of chocolates to each other.
Since They are giving each other an equal number of chocolates, The initial ratio and ratio after the first cycle will be the same.

I hope this helps!