On January 1st, 2023, a person saved ₹ 1. On January 2nd, 2023, he saved ₹ 2 more than that on the previous day. On January 3rd, 2023, he saved ₹ 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) 7th January, 2023
- (b) 8th January, 2023 ✓ UPSC's answer
- (c) 9th January, 2023
- (d) Not possible
Why the answer is (b)
• Daily savings are ₹1, ₹3, ₹5, … — the odd numbers. The total after n days is 1 + 3 + … + (2n − 1) = n².
• So the total is always a perfect square. It is also a perfect cube when n² is a cube, i.e. when n itself is a cube.
• The first cube after 1 is n = 8: total = 8² = 64 = 4³.
• Day 8 is 8 January 2023, option (b).
Why the other options are wrong
- (a) 7th January, 2023
- At the end of 7 January the total is 7² = 49, a square but not a cube.
- (c) 9th January, 2023
- At the end of 9 January the total is 9² = 81, a square but not a cube.
- (d) Not possible
- It is possible: on 8 January the total is 64 = 8² = 4³.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.