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# Kanika was given her pocket money on Jan $ 1^{\text {st }}, 2008 $. She puts Re 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?

Given:

Kanika was given her pocket money on Jan \( 1^{\text {st }}, 2008 \). She puts Re 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.

To do:

We have to find her pocket money for the month.

Solution:

Let her pocket money for the month be Rs. $x$.

She takes Rs. 1 on day 1, Rs. 2 on day 2, Rs. 3 on day 3 and so on till the end of the month, from this money.

This implies,

Total money taken $=1+2+3+4+\ldots+31$

This forms an AP.

Here,

First term $(a)=1$

Common difference $(d)=2-1=1$

We know that,

Sum of $n$ terms of an AP $S_{n}=\frac{n}{2}[2 a+(n-1) d]$

Sum of the amount taken $=S_{31}$

$S_{31}=\frac{31}{2}[2 \times 1+(31-1) \times 1]$

$=\frac{31}{2}(2+30)$

$=\frac{31 \times 32}{2}$

$=31 \times 16$

$=496$

Therefore,

Kanika takes Rs. 496 till the end of the month from the total money.

She spent Rs. 204 of her pocket money and found that at the end of the month she still has Rs. 100 with her.

This implies,

$(x-496)-204=100$

$x-700=100$

$x= 800$

**Hence, her pocket money for the month was Rs. 800.**

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