If the mean of the first n natural numbers is 15, what is the value of n? (2019)

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Question: If the mean of the first n natural numbers is 15, what is the value of n? (2019)

Options:

Correct Answer: 30

Exam Year: 2019

Solution:

Mean = n(n + 1) / (2n) = (n + 1) / 2 = 15. Therefore, n + 1 = 30, so n = 29.

If the mean of the first n natural numbers is 15, what is the value of n? (2019)

If the mean of the first n natural numbers is 15, what is the value of n? (2019)

Step 1: Understand that the first n natural numbers are 1, 2, 3, ..., n.
Step 2: The formula for the mean (average) of these numbers is: Mean = (Sum of numbers) / (Total numbers).
Step 3: The sum of the first n natural numbers is given by the formula: Sum = n(n + 1) / 2.
Step 4: The total number of natural numbers is n.
Step 5: So, the mean can be expressed as: Mean = (n(n + 1) / 2) / n.
Step 6: Simplify this to: Mean = (n + 1) / 2.
Step 7: We know from the question that the mean is 15, so we set up the equation: (n + 1) / 2 = 15.
Step 8: To eliminate the fraction, multiply both sides of the equation by 2: n + 1 = 30.
Step 9: Now, solve for n by subtracting 1 from both sides: n = 30 - 1.
Step 10: Therefore, n = 29.

Mean of Natural Numbers – Understanding how to calculate the mean of the first n natural numbers using the formula for the sum of an arithmetic series.
Algebraic Manipulation – Ability to manipulate equations to isolate variables and solve for n.