Question

There are seven consecutive integers. The average of the first five is $n$. What is the average of all seven?

• A. $n$
• B. $n+1$
• C. $kn$, here $k$ is a function of $n$
• D. $n + \frac{2}{7}$

Hint:

For consecutive integers, the average is the middle term. Shifting the range by $k$ changes the average by $k$.

Answer 1

The Correct Option is B

Let the seven consecutive integers be: \[ a,\ a+1,\ a+2,\ a+3,\ a+4,\ a+5,\ a+6 \] Average of first five: \[ \frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \frac{5a + 10}{5} = a + 2 \] So $n = a+2 \ \Rightarrow \ a = n - 2$. Average of all seven: \[ \frac{7a + 21}{7} = a + 3 = (n-2) + 3 = n+1 \] \[ \boxed{n+1} \]