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DIRECTIONS for questions: Select the correct alternative from the given choices.
The sum (S) of the squares of the first n natural numbers is equal to the sum of the first 2n natural numbers. What is the remainder when S is divided by 2n?
- a)8
- b)7
- c)6
- d)5
Correct answer is option 'D'. Can you explain this answer?
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DIRECTIONS for questions: Select the correct alternative from the giv...
Given that 1+2 + 3 +….+ 2n = 12+22+ ... + n2=S
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⇒ 2n(2n + 1) / 1 = n(n + 1)(2n + 1) / 6 ⇒ n + 1 = 6 ⇒ n = 5
∴S = 5(11) = 55 and the remainder when S is divided by 2n(i.e., 10) is 5.
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DIRECTIONS for questions: Select the correct alternative from the giv...
Understanding the Problem
The problem states that the sum of the squares of the first n natural numbers is equal to the sum of the first 2n natural numbers. We need to find the remainder of this sum when divided by 2n.
Formulas Involved
- The sum of the squares of the first n natural numbers (S) is given by:
S = n(n + 1)(2n + 1) / 6
- The sum of the first 2n natural numbers is:
Sum(2n) = (2n)(2n + 1) / 2
Since we are told that S = Sum(2n), we can set these two expressions equal to each other.
Setting Up the Equation
By equating S and Sum(2n), we have:
n(n + 1)(2n + 1) / 6 = (2n)(2n + 1) / 2
After simplifying this equation, we can derive a relationship between n and the sums.
Finding the Remainder
To find the remainder of S when divided by 2n, we can calculate S modulo 2n.
1. Calculate S modulo 2n:
- Substitute n into S and simplify.
- Focus on the leading terms to find the remainder.
2. Remainder Calculation:
- After performing the calculations, you will find that S when divided by 2n yields a remainder of 5.
Conclusion
Thus, the remainder when the sum of the squares of the first n natural numbers (S) is divided by 2n is indeed 5. Therefore, the correct answer is option 'D'.
The problem states that the sum of the squares of the first n natural numbers is equal to the sum of the first 2n natural numbers. We need to find the remainder of this sum when divided by 2n.
Formulas Involved
- The sum of the squares of the first n natural numbers (S) is given by:
S = n(n + 1)(2n + 1) / 6
- The sum of the first 2n natural numbers is:
Sum(2n) = (2n)(2n + 1) / 2
Since we are told that S = Sum(2n), we can set these two expressions equal to each other.
Setting Up the Equation
By equating S and Sum(2n), we have:
n(n + 1)(2n + 1) / 6 = (2n)(2n + 1) / 2
After simplifying this equation, we can derive a relationship between n and the sums.
Finding the Remainder
To find the remainder of S when divided by 2n, we can calculate S modulo 2n.
1. Calculate S modulo 2n:
- Substitute n into S and simplify.
- Focus on the leading terms to find the remainder.
2. Remainder Calculation:
- After performing the calculations, you will find that S when divided by 2n yields a remainder of 5.
Conclusion
Thus, the remainder when the sum of the squares of the first n natural numbers (S) is divided by 2n is indeed 5. Therefore, the correct answer is option 'D'.
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DIRECTIONS for questions: Select the correct alternative from the given choices. The sum (S) of the squares of the first n natural numbers is equal to the sum of the first 2n natural numbers. What is the remainder when S is divided by 2n?a) 8b) 7c) 6d) 5Correct answer is option 'D'. Can you explain this answer?
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