Q. On January 1, 2023, a person saved Rs. 1.

On January 2, 2003, he saved Rs. 2 more than that on the previous day.

On January 3, 2023, he saved Rs. 2 more than that on the previous day and so on.

# At the end of which date was his total savings a perfect square as well a perfect cube?

a) 7 January 2023

b) 8 January, 2023

c) 9 January, 2023

d) Not possible

Correct answer: b) 8 January, 2023

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

## Calculating final savings

Let’s approach this step-by-step:

1) First, let’s calculate the savings for each day:

Jan 1: 1

Jan 2: 1 + 2 = 3

Jan 3: 3 + 2 = 5

Jan 4: 5 + 2 = 7

Jan 5: 7 + 2 = 9

Jan 6: 9 + 2 = 11

Jan 7: 11 + 2 = 13

Jan 8: 13 + 2 = 15

Jan 9: 15 + 2 = 17

2) Now, let’s calculate the total savings for each day:

Jan 1: 1

Jan 2: 1 + 3 = 4

Jan 3: 4 + 5 = 9

Jan 4: 9 + 7 = 16

Jan 5: 16 + 9 = 25

Jan 6: 25 + 11 = 36

Jan 7: 36 + 13 = 49

Jan 8: 49 + 15 = 64

Jan 9: 64 + 17 = 81

3) We’re looking for a number that is both a perfect square and a perfect cube.

4) From our list, we can see that 64 (on Jan 8) is both a perfect square (8^2) and a perfect cube (4^3).

Therefore, the correct answer is b) 8 January, 2023.