What is the arithmetic mean of 8²,9²,10²,...,15²?

Calculation:

Given,

The sequence is \(8^{2},\,9^{2},\,10^{2},\,\dots,\,15^{2}\)

Number of terms, \(n = 15 - 8 + 1 = 8\)

The sum of the first \(m\) squares is  \( \displaystyle \sum_{k=1}^{m} k^{2} = \frac{m(m+1)(2m+1)}{6}\)

Sum up to \(15^{2}\):

\(S_{15} = \frac{15 \times 16 \times 31}{6} = 1240\)

Sum up to \(7^{2}\):

\(S_{7} = \frac{7 \times 8 \times 15}{6} = 140\)

Sum of the required terms:

\(S = S_{15} - S_{7} = 1240 - 140 = 1100\)

The arithmetic mean is,

\( \displaystyle \text{Mean} = \frac{S}{n} = \frac{1100}{8} = 137.5\)

∴ The arithmetic mean is \(137.5\).

Hence, the correct answer is option 3.