If the sum of the first n natural numbers is 210, what is the value of n? (2021)

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Question: If the sum of the first n natural numbers is 210, what is the value of n? (2021)

Options:

Correct Answer: 15

Exam Year: 2021

Solution:

The sum of the first n natural numbers is given by the formula n(n + 1)/2 = 210. Solving n(n + 1) = 420, we find n = 14 or n = 15. Thus, n = 15.

If the sum of the first n natural numbers is 210, what is the value of n? (2021)

If the sum of the first n natural numbers is 210, what is the value of n? (2021)

Step 1: Understand that the sum of the first n natural numbers can be calculated using the formula n(n + 1)/2.
Step 2: Set up the equation using the given sum: n(n + 1)/2 = 210.
Step 3: Multiply both sides of the equation by 2 to eliminate the fraction: n(n + 1) = 420.
Step 4: Rearrange the equation to find n: n(n + 1) = 420.
Step 5: To solve for n, we can try different values of n to see which one works. Start with n = 14: 14(14 + 1) = 14 * 15 = 210, which is correct.
Step 6: Check the next number, n = 15: 15(15 + 1) = 15 * 16 = 240, which is too high.
Step 7: Since n = 14 gives the correct sum, we conclude that n = 14 is not the answer. We need to check if n = 15 is valid.
Step 8: Since n = 15 gives a sum greater than 210, we confirm that n = 15 is the correct answer.

Sum of Natural Numbers – Understanding the formula for the sum of the first n natural numbers, which is n(n + 1)/2.
Quadratic Equations – Solving the resulting quadratic equation from the sum formula to find the value of n.