UPSC Prelims 2024 · CSAT
Question 10: Basic Numeracy
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?
AnswerUPSC’s answer is (b).
Daily savings are 1, 3, 5, and so on (odd numbers), so the total after n days is n². That is always a perfect square. It is also a perfect cube only when n² is a cube, i.e. when n itself is a perfect cube. Among 7, 8 and 9 only 8 = 2³ works: the total is 64 = 8² = 4³. So the total is both at the end of 8 January.
Why each option is right or wrong
- (a) Wrong. After 7 days the total is 49, a square but not a cube.
- (b) Right. After 8 days the total is 64 = 8² = 4³.
- (c) Wrong. After 9 days the total is 81, a square but not a cube.
- (d) Wrong. It is possible, on the 8th.
Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.
Source: UPSC, Civil Services (Preliminary) Examination 2024, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.