If the mean of 5 observations $x$, $x + 2$, $x + 4$, $x + 6$ and $x + 8$ is 11, then the mean of the last three observations is

### Concept & Strategy When a sequence of numbers increases by a constant amount, it forms an Arithmetic Progression (AP). The most powerful property of an AP with an odd number of terms is that its mean (average) is exactly equal to its middle term. $$Mean\ of\ AP = Middle\ Observation$$ ### Step-by-Step Solution * **Given:** The 5 observations are $x$, $x + 2$, $x + 4$, $x + 6$, $x + 8$. * The mean of these 5 observations is 11. * **Calculation:** Verify the sequence is an AP. The difference between consecutive terms is consistently 2. * The middle term of these 5 observations is the 3rd term: $x + 4$. * Because it is an AP, the mean equals the middle term. * $x + 4 = 11$ * We need the mean of the last three observations: $x + 4$, $x + 6$, $x + 8$. * This subset is also an AP. Its middle term is $x + 6$. * Since we know $x + 4 = 11$, we can find $x + 6$ by simply adding 2. * $x + 6 = 11 + 2 = 13$. ### Exam Strategy & Shortcut You do not need to find the value of $x$. The mean of the 5 terms is $x+4$. The mean of the last 3 terms is $x+6$. The difference between these means is exactly 2. So, $11 + 2 = 13$. This bypasses algebraic solving entirely. ### Common Pitfall The slow approach is calculating the full sum: $(5x + 20) / 5 = 11$, solving for $x = 7$, then substituting $x$ into the last three terms $(11, 13, 15)$, summing them (39), and dividing by 3 to get 13. While correct, this wastes massive amounts of time on a competitive exam. ### Final Answer **Therefore, the correct answer is 13.**