The mean of 9 observations is 16. What must a tenth observation be so that the mean of all 10 observations becomes 17?

The nine observations total $9\times 16=144$, and ten observations with mean 17 must total $10\times 17=170$, so the tenth observation is 170 − 144 = 26. The new value has to supply its own share of 17 PLUS the $9\times 1=9$ needed to lift each of the other nine by one unit, which is why the answer is 26 and not 18.