2070.82 – Mystery Function

Of course we can just plug the first few integers into the five given formulas and see which works, but here is a more subtle way.

We start by calculating some values:

$$\begin{aligned}f(1) &= 2, && \\f(2) &= 2 + 2 &&= 4, \\f(3) &= 4 + 3 &&= 7, \\f(4) &= 7 + 4 &&= 11, \\f(5) &= 11 + 5 &&= 16.\end{aligned}$$

With each step we are adding one more number so $f$ should be something like $f(n) \approx 1 + 2 +3 \ldots + n$. Let's calculate this sum:

$$\begin{aligned}1, & \\1 + 2 &= 3, \\1 + 2 + 3 &= 6, \\1 + 2 + 3 + 4 &= 10, \\1 + 2 + 3 + 4 + 5 &= 15. \\\end{aligned}$$

So, we see that

$$f(n) = (1 + 2 + \ldots + n) + 1 = \dfrac{n(n+1)}{2} + 1 = \dfrac{n^2 + n + 2}{2}.$$

The correct answer is (c). (That formula for the sum of the first n whole numbers is a handy one to know.)

For inquiring minds: what if $f(1)$ isn't $2$? What if $f(1)$ is, say, $3$? or $f(1) = n$?