If f(n)=13n(n+1)(n+2), then f(r)-f(r-1) equals:

1968 AMC 12 Problem 5

Attempt Problem 5 of the 1968 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 1968 AMC 12 solutions, or check the answer key.

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f(n)=\dfrac13n(n+1)(n+2),

then

f(r)-f(r-1)

equals:

r(r+1)

(r+1)(r+2)

\dfrac13r(r+1)

\dfrac13(r+1)(r+2)

\dfrac13r(r+1)(r+2)

Substitute

r

and

r-1

separately

Factor out

\frac13r(r+1)

We have

f(r)=\frac13r(r+1)(r+2)

and

f(r-1)=\frac13(r-1)r(r+1).

Factor their difference as

\frac13r(r+1)

times

(r+2)-(r-1)=3.

The result is

r(r+1).

Therefore, the correct answer is A.