The average of 5 consecutive numbers, in increasing order is $ n $. If the next three numbers are also include

Question

The average of 5 consecutive numbers, in increasing order is $ n $. If the next three numbers are also included, the average will:

Examoogle · Accepted Answer

The correct answer is Option D: increase by 1.5. For a sequence of consecutive integers, adding $k$ consecutive numbers after the original set increases the average by $k/2$. Here $k=3$, so the average increases by $3/2 = 1.5$.

Step-by-Step Solution & Concepts:
Shortcut Trick: For a sequence of consecutive integers, adding $k$ consecutive numbers after the original set increases the average by $k/2$. Here $k=3$, so the average increases by $3/2 = 1.5$.

Formula Used: $$Average = \frac{\sum x_i}{k}$$

Given: Average of 5 consecutive numbers is n. The next 3 consecutive numbers are added.

Steps: Define initial sequence: Let the 5 consecutive numbers be $x, x+1, x+2, x+3, x+4$. Their sum is $5x + 10$. The average $n$ is $\frac{5x+10}{5} = x+2$.. Add next three numbers: The next three numbers are $x+5, x+6, x+7$. The total sum for the 8 numbers is $(5x + 10) + (x+5 + x+6 + x+7) = 8x + 28$.. Calculate new average: The new average of the 8 numbers is $\frac{8x+28}{8} = x + 3.5$.. Determine increase: The difference between the new average and the old average is $(x + 3.5) - (x + 2) = 1.5$.

Answer: D is correct because the average increases from $x+2$ to $x+3.5$, resulting in an increase of 1.5.

Exam Tip: This concept is a specific case of the property: for an arithmetic progression of $k$ terms, adding $m$ consecutive terms shifts the average by $\frac{m}{2} 	imes d$, where $d$ is the common difference.

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