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__________ is both a triangular number and a square number.
- a)25
- b)36
- c)8
- d)4
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
__________ is both a triangular number and a square number.a)25b)36c)8...
- A triangular number is a number that can form an equilateral triangle. The sequence of triangular numbers is: 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
- A square number is a number that is the square of an integer. The sequence of square numbers is: 1, 4, 9, 16, 25, 36, 49, 64, ...
The number 36 appears in both sequences:
- It is the 8th triangular number: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.
- It is also the 6th square number: 6 * 6 = 36.
Thus, 36 is both a triangular number and a square number.
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__________ is both a triangular number and a square number.a)25b)36c)8...
Understanding Triangular Numbers
Triangular numbers are formed by the sum of the natural numbers. The nth triangular number is given by the formula:
- T(n) = n(n + 1)/2
For example:
- T(1) = 1
- T(2) = 3
- T(3) = 6
- T(4) = 10
- T(5) = 15
- T(6) = 21
- T(7) = 28
- T(8) = 36
Understanding Square Numbers
Square numbers are the result of multiplying a number by itself. The nth square number is given by the formula:
- S(n) = n^2
For example:
- S(1) = 1
- S(2) = 4
- S(3) = 9
- S(4) = 16
- S(5) = 25
- S(6) = 36
Identifying the Correct Option
Now, let's evaluate the options to see which one is both a triangular and a square number.
- Option a) 25: Not a triangular number (T(6) = 21, T(7) = 28)
- Option b) 36:
- Triangular: T(8) = 36
- Square: S(6) = 36
- Option c) 8: Not a triangular or square number
- Option d) 4: Square (S(2) = 4) but not triangular (T(2) = 3)
Conclusion
The only number that is both a triangular number and a square number is 36. Hence, the correct answer is option b.
Triangular numbers are formed by the sum of the natural numbers. The nth triangular number is given by the formula:
- T(n) = n(n + 1)/2
For example:
- T(1) = 1
- T(2) = 3
- T(3) = 6
- T(4) = 10
- T(5) = 15
- T(6) = 21
- T(7) = 28
- T(8) = 36
Understanding Square Numbers
Square numbers are the result of multiplying a number by itself. The nth square number is given by the formula:
- S(n) = n^2
For example:
- S(1) = 1
- S(2) = 4
- S(3) = 9
- S(4) = 16
- S(5) = 25
- S(6) = 36
Identifying the Correct Option
Now, let's evaluate the options to see which one is both a triangular and a square number.
- Option a) 25: Not a triangular number (T(6) = 21, T(7) = 28)
- Option b) 36:
- Triangular: T(8) = 36
- Square: S(6) = 36
- Option c) 8: Not a triangular or square number
- Option d) 4: Square (S(2) = 4) but not triangular (T(2) = 3)
Conclusion
The only number that is both a triangular number and a square number is 36. Hence, the correct answer is option b.
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