If the sum of the first n natural numbers is 5050, what is the value of n? (2021

If the sum of the first n natural numbers is 5050, what is the value of n? (2021)

If the sum of the first n natural numbers is 5050, 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 = 5050.
Step 3: To eliminate the fraction, multiply both sides of the equation by 2: n(n + 1) = 10100.
Step 4: Now, we need to solve the equation n(n + 1) = 10100. This means we need to find two consecutive numbers that multiply to 10100.
Step 5: Rewrite the equation as n^2 + n - 10100 = 0. This is a quadratic equation.
Step 6: Use the quadratic formula n = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = 1, and c = -10100.
Step 7: Calculate the discriminant: b² - 4ac = 1² - 4(1)(-10100) = 1 + 40400 = 40401.
Step 8: Find the square root of the discriminant: √40401 = 201.
Step 9: Substitute back into the quadratic formula: n = (-1 ± 201) / 2.
Step 10: Calculate the two possible values for n: n = (200) / 2 = 100 and n = (-202) / 2 (which is negative and not valid).
Step 11: Conclude that the only valid solution is n = 100.

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