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?

#### Solution

Let her pocket money be ₹ x

Now, she puts 11 on day 1, ₹ 2 on day 2, ₹ 3 on day 3

And so on till the end of the month, from this money into her piggy bank.

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 is S_{31}

Sum of n terms, `S_n = n/2[2a + (n - 1)d]`

∴ `S_31 = 31/2[2 xx 1 + (31 - 1) xx 1]`

= `31/2(2 + 30)`

= `(31 xx 32)/2`

= `331 xx 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

⇒ x – 700 = 100

∴ x = ₹ 800

Hence, ₹ 800 was her pocket money for the month.