What is the average of the first seven positive multiples of 7 (that is, 7, 14, 21, 28, 35, 42 and 49)?

Introduction / Context:
This question checks your understanding of averages of equally spaced numbers, specifically an arithmetic progression. The first seven positive multiples of 7 form a simple arithmetic sequence with constant difference. Recognizing this pattern allows you to compute the average efficiently without adding every term individually, a useful shortcut for timed exams.

Given Data / Assumptions:

• The sequence is 7, 14, 21, 28, 35, 42 and 49.
• Number of terms = 7.
• This is an arithmetic progression with first term 7 and last term 49.
• Common difference = 7.

Concept / Approach:
For any arithmetic progression, the average (arithmetic mean) of all terms is simply equal to the mean of the first and last term. That is, average = (first term + last term) / 2. This is because the terms are symmetrically spaced around the middle value, and each pair equidistant from the ends sums to the same constant.

Step-by-Step Solution:
Step 1: Identify first term a = 7 and last term l = 49.Step 2: Use the property of arithmetic progressions: average = (a + l) / 2.Step 3: Compute (7 + 49) / 2 = 56 / 2.Step 4: 56 / 2 = 28.Step 5: Therefore, the average of the first seven multiples of 7 is 28.

Verification / Alternative check:
We can also directly sum the seven terms and divide by 7. Sum = 7 + 14 + 21 + 28 + 35 + 42 + 49 = 196. Then average = 196 / 7 = 28. Both methods give the same result, confirming the calculation is correct.

Why Other Options Are Wrong:
Option 7 is just the first term, not the average. Option 14 is the second term, and option 21 is the fourth term, both below the true central value. Option 35 is above the true average and would shift the balance of total too high. Only 28 matches both the arithmetic progression property and the explicit computation.

Common Pitfalls:

• Assuming the average is equal to the middle term without confirming the sequence is an arithmetic progression.
• Making arithmetic errors while adding several terms under exam pressure.
• Confusing the concept of multiples with that of powers.

Final Answer:
The average of the first seven multiples of 7 is 28.