Bunuel wrote:

If the average (arithmetic mean) of seven consecutive integers is k + 2, then the product of the greatest and least integer is

A. k^2 - 9

B. k^2 - 2k + 1

C. k^2 + 4k - 5

D. k^2 + 6k + 9

E. k^2 + 4k - 12

I used algebra, erred, and took a long time to correct. So I timed myself using what I thought was a time-consuming method. Wrong. I was well under a minute.

I used only one sketch, and no fancy lines or asterisks. I used them here for clarity.

1. List the integers

x|x+1|x+2|x+3|x+4|x+5|x+6|

2. Place (k+2) under the median

"The arithmetic mean is k + 2." Median = mean for evenly spaced set. So

x|x+1|x+2|x+3|x+4|x+5|x+6|

*|***|***|k+2|***|***|***|

3. Find k. From above:

k + 2 = x + 3

k = x + 1, so

x|x+1|x+2|x+3|x+4|x+5|x+6|

*|_k_|***|k+2|***|***|***|

4. Find answer. The product of the greatest and least integer is? We need x, and (x+6), in terms of k, so

_x_|x+1|x+2|x+3|x+4|x+5|x+6|

k-1|_k_ |***|k+2|***|***|k+5|

(k - 1)(k + 5) = k\(^2\) + 4k - 5

ANSWER C

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"