**Question:**

Kanika was given her pocket money on Jan 1st, 2008. She puts ₹ 1 on day 1, ₹ 2 on day 2, ₹ 3 on day 3 and continued doing so till the end of the

month, from this money into her piggy bank she also spent ₹ 204 of her pocket money, and found that at the end of the month she still had ₹ 100 with

her. How much was her pocket money for the month?

**Solution:**

Let her pocket money be ₹ x.

Now, she takes ₹ 1 on day 1, ₹ 2 on day 2, ₹ 3 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 = S31

$\therefore$ Sum of first 31 terms $=S_{31}$

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

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

$=\frac{31}{2}(2+30)=\frac{31 \times 32}{2}$

$=31 \times 16=496$

So, Kanika takes ₹ 496 till the end of the month from this money.

Also, she spent ₹ 204 of her pocket money and found that at the end of the month she still has ₹ 100 with her.

Now, according to the condition,

$(x-496)-204=100$

$\therefore \quad x=₹ 800$

Hence, ₹ 800 was her poket money for the month

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