Average of First 167 Even Numbers Calculation
To find the average of the first 167 even numbers, we can use a known property of arithmetic sequences.
The first $n$ positive even numbers are $2, 4, 6, \ldots, 2n$. The average of the first $n$ even numbers is given by the formula: $ \text{Average} = \frac{\text{Sum of first } n \text{ even numbers}}{n} $ The sum of the first $n$ even numbers is $n(n+1)$. So, the average is:
$ \text{Average} = \frac{n(n+1)}{n} = n+1 $In this problem, we need to find the average of the first 167 even numbers, so $n=167$. Using the formula:
$ \text{Average} = n+1 = 167 + 1 = 168 $Therefore, the average of the first 167 even numbers is 168.
Steps Summary:
- Identify the number of terms, $n = 167$.
- Apply the formula for the average of the first $n$ even numbers: Average = $n+1$.
- Calculate the average: $167 + 1 = 168$.
The correct option is 168.
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