jmc

Question

For all real numbers r and s, define the mathematical operation # such that the following conditions apply: r # 0 = r, r # s = s # r, and (r + 1) # s = (r # s) + s + 1. What is the value of 11 # 5?

Accepted Answer

Using the first two conditions, we have that

0#11 = 11#0 = 11.

Using the third condition, with

r = 0

and

s = 11

, we have that

1#11 = (0#11) + 12 = 11 + 12.

As we increase

r

by

1

, we increase

r #11

by

s + 1 = 11 + 1 = 12

. Since we want to increase

r

5

times to find

11#5 = 5#11

, we want to increase

0#11

by

12

five times. Therefore, we have

11#5 = 5#11 = 11 + 5 \cdot 12 = 11 + 60 = 71 .

More generally,

a # b = ab + a + b .