Question

Rahul's father wants to deposit some amount of money every year on the day of Rahul's birthday. On his $1st$ birth day $Rs.100$, on his $2nd$ birth day $Rs.300$, on his $3rd$ birth day $Rs.600$, on his $4th$ birthday $Rs.1000$ and so on.

What is the amount deposited by his father on Rahul's $15th$ birthday.

Answer 1

Let $T_n$ be the $n^{th}$ term of the series, then

$S=100[1+3+6+10+15........T(n-1)+T(n\left)\right]$

$S=100[1+3+6+\mathrm{10.....................}+T(n-1)+T(n\left)\right]$

Subtract both equations :

$0=100[1+(2+3+4+5.......T\left(n\right)-T(n-1))-T(n\left)\right]$

$T\left(n\right)=100[1+2+3+4+5.........n]$ (This is the series of the difference between yearly deposits]

It can be seen that the difference between the year's amounts are

$2nd-1st=300-100=200$

$3rd-2nd=600-300=300$

$4th-3rd=1000-600=400$)

$T\left(n\right)=100\left[\frac{n(n+1)}{2}\right]$

Now we can tell by this formula of T(n),The amount deposited on $15th$ Birthday as follows

$T\left(15\right)=100\times \frac{15\times 16}{2}=12000$

Therefore on his $15th$ birthday his father will deposit $12000$.