The mean of 3, 4, 9, 2k, 10, 8, 6 and (k + 6) is 8. So the sum of these numbers divided by 8 is 8.
\(\frac{3 + 4 + 9 + 2k + 10 + 8 + 6 + (k+6)}{8} = 8\)
\(\frac{46 + 3k}{8} = 8\)
\(46 + 3k = 64\)
\(3k = 18\)
\(k = 6\)
The mode of 2, 2, 3, 2p, (2p + 1), 4, 4, 5 and 6 is 4, meaning 4 appears most often.
Since 2,3,5,6 only appear once, either 2p or 2p+1 needs to be 4 in order for 4 to be the mode.
If 2p=4, then p=2. If 2p+1=4, this will lead to p=3/2 which is not a natural number.
Thus, the mode equals 4 when p=2 Now we calculate k - 2p = 6 - 2 \(\times\) 2 = 6 - 4 = 2