Question

The average of 8 consecutive even numbers is 35. The largest number is:

• A. 40
• B. 42
• C. 44
• D. 46

Hint:

For an even count of consecutive elements, the average is the midpoint of the two central numbers. Here, 35 sits exactly between the 4th and 5th even numbers, which must be 34 and 36. From 36 (the 5th number), just count up by two three more times to get the 8th (largest) number: $36 \rightarrow 38 \rightarrow 40 \rightarrow 42$. No algebra needed!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For any arithmetic progression (such as consecutive even numbers), the average is always located exactly at the geometric midpoint of the sequence. If there is an even number of terms (like 8), the average will be the exact midpoint between the two middle numbers (the 4th and 5th numbers).

Step 2: Key Formula or Approach:
Let the 8 consecutive even numbers be represented algebraically as: $$n, n+2, n+4, n+6, n+8, n+10, n+12, n+14$$ $$\text{Average} = \frac{\text{Sum of all terms}}{\text{Total number of terms}}$$

Step 3: Detailed Explanation:
Let's find the sum of our algebraic representation: \[ \text{Sum} = n + (n+2) + (n+4) + (n+6) + (n+8) + (n+10) + (n+12) + (n+14) \] \[ \text{Sum} = 8n + 56 \] Given that the average of these 8 numbers is 35: \[ \text{Average} = \frac{8n + 56}{8} = 35 \] \[ n + 7 = 35 \] \[ n = 35 - 7 = 28 \] The first (smallest) even number in this sequence is $28$. Now, we calculate the largest number in the sequence, which is represented by $(n + 14)$: \[ \text{Largest Number} = n + 14 = 28 + 14 = 42 \]

Step 4: Final Answer:
The largest consecutive even number in the series is 42.