1997 AMC 12 Problem 14

Attempt Problem 14 of the 1997 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

The number of geese in a flock increases so that the difference between the populations in year

n+2

and year

n

is directly proportional to the population in year

n+1.

If the populations in the years

1994,

1995,

and

1997

were

39,

60,

and

123,

respectively, then the population in

1996

was

81

84

87

90

102

Let

x

be the

1996

population and

k

the constant of proportionality

Translate the years

1994

–

1996

and

1995

–

1997

into two equations using the same

k

The rule gives

x-39=60k

and

123-60=kx.

Eliminating

k

yields

x(x-39)=3780,

(x-84)(x+45)=0.

The population is positive, so

x=84,

and B is correct.

Problem 14 in other years