If the mean of the first n odd natural numbers is n²/81, then n = ?

Solution: Find n such that the mean of the first n odd natural numbers equals n²/81.

• Step 1: Sum and mean formula. The sum of the first n odd numbers is n², so the mean is n²/n = n.

• Step 2: Set the mean equal to the given value: n = n²/81. Rearranging gives n² - 81n = 0, i.e., n(n - 81) = 0.

• Step 3: Solve: n = 0 or n = 81. Since n is a positive natural number, n = 81.

• Check: For n = 81, the mean is 81²/81 = 81, which matches the mean computed from the sum (n).