Kanika was given her pocket money on Jan `1st` , `2008`. She puts `Rs.` `1` on day `1`, `Rs.` `2` on day `2`, `Rs.` `3` on day `3` and continued doing so till the end of the month, from this money into her piggy bank she also spent `Rs.` `204` of her pocket money, and found that at the end of the month she still had `Rs.` `100` with her. How much was her pocket money for the month ?

Explanation

Let her pocket money be Rs. `x`.
Now, she takes Rs. 1 on day 1, Rs. 2 on day 3 and so on till the end of the month, from this money.
i.e., `1+2+3+4+ … +31`
which form an AP in which terms are 31 and first term (a) = 1, common difference (d) `= 2-1=1`
` :. ` Sum of first 31 terms `=S_(31)`
Sum of n terms,
`S_(n)= (n)/(2)[2a+(n-1)d]`
`:. S_(n)= (31)/(2)[2xx1+(31-1)xx1]`
` " " =(31)/(2)(2+30)=(31xx32)/(2)`
` " " =31xx16=496`
So, Kanika takes Rs. 496 till the end of the month from this money.
Also, she spent Rs. 204 of her pocket money and found that at the end of the month she still has Rs. 100 with her.
Now, according to the condition,
` (x-496)-204=100`
` implies x-700=100`
` :. x=Rs. 800`
Hence, Rs. 800 was her pocket money for the month.