Q10. On January 1st, 2023, a person saved Rs 1. On January 2nd, 2023, he saved Rs 2 more than that on the previous day. On January 3rd, 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 as a perfect cube?
A. 7th January, 2023
B. 8th January, 2023
C. 9th January, 2023
D. Not possible
Answer: B. 8th January, 2023
Explanation:
The person’s savings increase by Rs. 2 each day. This forms a sequence of daily savings: 1, 3, 5, 7, 9, 11, 13, 15, ...
Let’s calculate the total savings day by day and check when it becomes both a perfect square and a perfect cube.
Calculations:
- 1st January: Savings = 1
Total savings = 1
(1 is both a perfect square and a perfect cube)
- 2nd January: Savings = 3
Total savings = 1 + 3 = 4
(4 is a perfect square but not a perfect cube)
- 3rd January: Savings = 5
Total savings = 1 + 3 + 5 = 9
(9 is a perfect square but not a perfect cube)
- 4th January: Savings = 7
Total savings = 1 + 3 + 5 + 7 = 16
(16 is a perfect square but not a perfect cube)
- 5th January: Savings = 9
Total savings = 1 + 3 + 5 + 7 + 9 = 25
(25 is a perfect square but not a perfect cube)
- 6th January: Savings = 11
Total savings = 1 + 3 + 5 + 7 + 9 + 11 = 36
(36 is a perfect square but not a perfect cube)
- 7th January: Savings = 13
Total savings = 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
(49 is a perfect square but not a perfect cube)
- 8th January: Savings = 15
Total savings = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64
(64 is both a perfect square and a perfect cube, as 64 = 82 = 43)
Thus, the total savings amount to 64 on 8th January, 2023, which is both a perfect square and a perfect cube.
Therefore, the correct answer is B. 8th January, 2023.
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