CSAT 2024

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.

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