Elephant Seals

[ZGirl1]

Okay, there's an island, and on the first day, there are 5 seals on the island. On the second day, 6 more seals arrive, so now there are 11 seals on the island. On the third day, 7 more seals arrive, so by day 3, there are 18 seals on the island. The pattern continues indefinitely. We are trying to figure out what the "algebraic rule" for this pattern would be. We can only seem to do it if we base it off of the previous answer. That doesn't help when we want to just plug in numbers to an algebraic equation to figure out how many seals will be there on day 100! Ideas anyone?

 

[MarkFL]

One way to derive this, and perhaps the simplest to see, is to let \(\displaystyle S_n\) be the number of seals present on day \(\displaystyle n\). We may then write:

\(\displaystyle S_n=5+6+7+\cdots+(n+2)+(n+3)+(n+4)\)

\(\displaystyle S_n=(n+4)+(n+3)+(n+2)+\cdots+7+6+5\)

Adding the two equations, we find:

\(\displaystyle 2S_n=n(n+9)\)

Do you see that we have \(\displaystyle n+9\) a total of \(\displaystyle n\) times?

Now, divide through by 2 to get:

\(\displaystyle S_n=\dfrac{n(n+9)}{2}\)