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 Question:
The topic of this problem is Averages (Mean) of Arithmetic Progressions. In this case, the series consists of "consecutive even numbers," which implies a sequence of numbers where each term is exactly 2 units greater than the preceding one. We are dealing with a set of 8 such numbers, and we know that their mathematical average is 35. The goal is to determine the highest value in this specific set.
Step 2 : Key Formulas and approach:
There are two primary ways to approach this: 1. Algebraic Method: Represent the numbers as $x, x+2, x+4, \dots, x+14$. The sum divided by the count equals the average.
2. Properties of Symmetry: In an arithmetic sequence with an even number of terms, the average is the exact midpoint between the two central terms.
The approach involves locating the "middle" of the sequence and counting forward to the last term.
Step 3 : Detailed Explanation:

First, we recognize that the sequence contains 8 terms. In a symmetric sequence, the average always sits at the geometric center. For 8 numbers, the center is the space between the 4th and the 5th number.

Since the average is 35, and we are looking for even numbers, 35 must be the exact midpoint between the 4th even number and the 5th even number.

The even integer immediately below 35 is 34 (the 4th number), and the even integer immediately above 35 is 36 (the 5th number). This confirms our sequence is balanced around 35.

Now, we can list the sequence by moving outward from these central points. If the 5th number is 36, then the 6th number is $36 + 2 = 38$.

Continuing this pattern, the 7th number is $38 + 2 = 40$.

Finally, the 8th and largest number is $40 + 2 = 42$. We can also verify the lower half: if the 4th is 34, then the 3rd is 32, the 2nd is 30, and the 1st is 28.

To verify algebraically: $\text{Sum} = 28+30+32+34+36+38+40+42 = 280$. $\text{Average} = \frac{280}{8} = 35$. The calculation is consistent with the given data.

Step 4 : Final Answer:
The largest even number in this sequence is 42, which matches option (B).