Question

Krishna has certain number of coins numbered with a series of consecutive natural numbers starting with 1. He found that the sum of squares of all the numbers on the coins is 1753 times the sum of numbers on coins. How many coins does he have?

• A. 1753
• B. 2569
• C. 2629
• D. 3439
• E. 5259

Answer 1

The Correct Option is C

Step 1: Understand the problem.
Krishna has a certain number of coins numbered with a series of consecutive natural numbers starting with 1. We are told that the sum of squares of all the numbers on the coins is 1753 times the sum of the numbers on the coins. We need to find how many coins Krishna has.

Step 2: Define the variables.
Let the number of coins be \( n \). The coins are numbered from 1 to \( n \). The two key sums are:
- The sum of the numbers on the coins is: \[ \text{Sum of numbers} = \frac{n(n + 1)}{2} \] - The sum of the squares of the numbers on the coins is: \[ \text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6} \]

Step 3: Set up the equation.
According to the problem, the sum of squares is 1753 times the sum of numbers: \[ \frac{n(n + 1)(2n + 1)}{6} = 1753 \times \frac{n(n + 1)}{2} \] Simplifying both sides: \[ \frac{n(n + 1)(2n + 1)}{6} = \frac{1753 \times n(n + 1)}{2} \] Canceling out \( n(n + 1) \) from both sides (since \( n \neq 0 \)): \[ \frac{2n + 1}{6} = \frac{1753}{2} \] Multiplying both sides by 6 to eliminate the denominator: \[ 2n + 1 = 1753 \times 3 = 5259 \] Solving for \( n \): \[ 2n = 5259 - 1 = 5258 \] \[ n = \frac{5258}{2} = 2629 \]

Step 4: Conclusion.
Krishna has 2629 coins.

Final Answer:
The correct option is (C): 2629.