Kanika deposits Rs 1 on day 1, Rs 2 on day 2, … through January 2008 (31 days). She also spent Rs 204 and still had Rs 100 left. What was her pocket money for the month?
Rs 800
Step 1: Kanika deposits money every day. On day 1 she puts Rs 1, on day 2 she puts Rs 2, … up to day 31 she puts Rs 31. This forms a series:
\(1 + 2 + 3 + \dots + 31\).
Step 2: To find the total of this series, we use the formula for the sum of the first \(n\) natural numbers:
\(S = \dfrac{n(n+1)}{2}\).
Step 3: Here, \(n = 31\). So,
\(S = \dfrac{31 \times (31+1)}{2} = \dfrac{31 \times 32}{2}\).
\(S = \dfrac{992}{2} = 496\).
This means Kanika saved Rs 496 in total.
Step 4: We are told she also spent Rs 204 and still had Rs 100 left with her at the end of the month.
Step 5: So, her total pocket money must be the sum of all three amounts:
Total pocket money = Savings + Amount spent + Amount left
= \(496 + 204 + 100\).
Step 6: Add them: \(496 + 204 = 700\). Then, \(700 + 100 = 800\).
Final Answer: Kanika’s pocket money for the month = Rs 800.