Can you count all these oranges?

When we separate this sequence by parts or layers of the oranges, we will see a pattern as shown:

$a_{1}$ = 1

$a_{2}$ = 1 + 3 = 1 + (1 + 2)

$a_{3}$ = 1 + 3 + 6 = 1 + (1 + 2) + (1 + 2 + 3)

...

$a_{n}$ = 1 + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + 3 + ... + n)

So we can see that this series is, in fact, a sum of sums of all natural numbers under n.

That is, $a_{n}$ = $\sum (\sum n$ ) = $\sum (\dfrac{n(n+1)}{2}$ ) as we know that the sum of 1 to n equals to $\dfrac{n(n+1)}{2}$ from the formula.

Now $a_{n}$ = $\sum (\dfrac{n(n+1)}{2}$ ) = $\dfrac{1}{2}$ [ $\sum n^2$ + $\sum n$ ]

From the formulas, $\sum n^2$ = $\dfrac{n(n+1)(2n+1)}{6}$ and again $\sum n$ = $\dfrac{n(n+1)}{2}$ , we will get:

$a_{n}$ = $\dfrac{1}{2}$ [ $\sum n^2$ + $\sum n$ ] = $\dfrac{1}{2}$ [ $\dfrac{n(n+1)(2n+1)}{6}$ + $\dfrac{n(n+1)}{2}$ ]

= $\dfrac{1}{2}$ [ $\dfrac{n(n+1)(2n+1+3)}{6}$ ] = $\dfrac{1}{2}$ [ $\dfrac{n(n+1)(2n+4)}{6}$ ] = $\dfrac{n(n+1)(n+2)}{6}$

Therefore, $a_{100}$ = $\dfrac{100(100+1)(100+2)}{6}) = 171,700$ .

As a result, there will be $\boxed{171,700}$ oranges when 100 layers of oranges are made.