1968 AHSME Problems/Problem 5

Problem

If $f(n)=\tfrac{1}{3} n(n+1)(n+2)$, then $f(r)-f(r-1)$ equals:

$\text{(A) } r(r+1)\quad \text{(B) } (r+1)(r+2)\quad \text{(C) } \tfrac{1}{3} r(r+1)\quad  \\ \text{(D) } \tfrac{1}{3} (r+1)(r+2)\quad \text{(E )} \tfrac{1}{3} r(r+1)(2r+1)$

Solution

Plugging in the expressions for $f(r)$ and $f(r-1)$, we see that: f(r)f(r1)=13r(r+1)(r+2)13(r1)(r1+1)(r1+2)=13[r(r+1)(r+2)r(r1)(r+1)]=13[r(r2+3r+2)r(r21)]=13[r3+3r2+2rr3+r]=13[3r2+3r]=r2+r=r(r+1), which is answer choice $\fbox{A}$.

See also

1968 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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