Determine the 288th term of the series
a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f ... :

The pattern:

a appears 1 time
b appears 2 times
c appears 3 times
d appears 4 times
e appears 5 times, and so on. 

Hence, we need cumulative sums. The sum of the first n numbers can be given by the formula: $$\dfrac{n\left(n+1\right)}{2}$$

We want to find the 288th term. 

Hence, $$\dfrac{n\left(n+1\right)}{2}\ge\ 288$$

Checking:

When n = 23, $$\dfrac{\left(23\times24\right)}{2}=276$$

When n = 24, $$\dfrac{\left(24\times25\right)}{2}=300$$

So, the 288th term lies in the 24th group

24th alphabet is x.