Calculating the Average of First 7 Non-Zero Multiples of 17

This question asks for the average of a specific set of numbers: the first 7 non-zero multiples of 17. To find the average, we need to identify these numbers, sum them up, and then divide by the count of the numbers.

Understanding Non-Zero Multiples

A multiple of 17 is a number that can be obtained by multiplying 17 by an integer. Non-zero multiples exclude the result of multiplying 17 by 0 (which is 0). So, the non-zero multiples of 17 are $17 \times 1$, $17 \times 2$, $17 \times 3$, and so on.

Identifying the First 7 Non-Zero Multiples of 17

The first 7 non-zero multiples of 17 are:

• $17 \times 1 = 17$
• $17 \times 2 = 34$
• $17 \times 3 = 51$
• $17 \times 4 = 68$
• $17 \times 5 = 85$
• $17 \times 6 = 102$
• $17 \times 7 = 119$

So, the set of numbers is $\{17, 34, 51, 68, 85, 102, 119\}$. There are 7 numbers in this set.

Method 1: Summing and Averaging the Multiples

The average of a set of numbers is calculated using the formula:

$$ \text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} $$

First, let's find the sum of these 7 multiples:

$$ \text{Sum} = 17 + 34 + 51 + 68 + 85 + 102 + 119 $$

We can factor out 17:

$$ \text{Sum} = 17 \times (1 + 2 + 3 + 4 + 5 + 6 + 7) $$

The sum of the first 7 natural numbers $(1+2+3+4+5+6+7)$ is $\frac{7 \times (7+1)}{2} = \frac{7 \times 8}{2} = \frac{56}{2} = 28$.

So, the sum of the multiples is:

$$ \text{Sum} = 17 \times 28 $$

Let's perform the multiplication:

The sum of the first 7 non-zero multiples of 17 is 476.

Now, we can calculate the average. The count of numbers is 7.

$$ \text{Average} = \frac{476}{7} $$

Performing the division:

The average of the first 7 non-zero multiples of 17 is 68.

Method 2: Using Properties of Arithmetic Progression

The first 7 non-zero multiples of 17 (17, 34, 51, 68, 85, 102, 119) form an arithmetic progression (AP). In an AP, the difference between consecutive terms is constant (in this case, the common difference is 17).

For an arithmetic progression with an odd number of terms, the average is equal to the middle term. There are 7 terms, so the middle term is the $(\frac{7+1}{2}) = 4\text{th}$ term.

The 4th term is the 4th non-zero multiple of 17, which is $17 \times 4 = 68$.

Alternatively, for any arithmetic progression, the average is also the average of the first and the last term.

• First term = 17 (1st multiple)
• Last term = 119 (7th multiple)

$$ \text{Average} = \frac{\text{First term} + \text{Last term}}{2} $$

$$ \text{Average} = \frac{17 + 119}{2} = \frac{136}{2} = 68 $$

Both methods confirm that the average of the first 7 non-zero multiples of 17 is 68.

Conclusion on Average Calculation

We have successfully calculated the average of the specified set of numbers using two different methods, both yielding the same result. The first 7 non-zero multiples of 17 are 17, 34, 51, 68, 85, 102, and 119. Their average is 68.

Revision Table: Key Concepts for Average of Multiples

Additional Information on Averages and Multiples

Understanding averages and multiples is fundamental in number theory and arithmetic. Multiples are essentially the results of multiplication tables for a given number. The concept of non-zero multiples is important when the context specifically excludes the number 0.

Averages, or means, are a measure of central tendency. They give us a single value that represents the center of a dataset. For evenly spaced sets of numbers, like multiples or numbers in an arithmetic progression, calculating the average is simpler because of their inherent pattern. The properties of arithmetic progressions, such as the average being the middle term (for odd count) or the average of the first and last terms, provide convenient shortcuts.

This type of problem often appears in competitive exams to test basic arithmetic skills and understanding of number properties. Practicing with different numbers and counts of multiples can help solidify the concept.