Explaining P(k)=c(101-k) for Finding 'c' in Turtle's Life Expectancy

[cdux]

That's not exactly coursework (it's studying) but I've spammed the probabilities forum too much.

Can someone explain this part?

Knowing that P(k)=P(X=k)=c(101-k)

For a 'turtle' that can live up to 100 years, find 'c'.

And it goes

I don't understand the jump from cΣk to c100(100+1)/2

 

[DrClaude]

$$
\sum_{k=1}^n k = (1+n) \frac{n}{2}
$$
To see how to get the formula, list the numbers of the sum, then sum the first with the last, the second with the second to last, etc.

 

[cdux]

then sum the first with the last, the second with the second to last, etc.

Ah, hacky. I was expecting a more straightforward way.

Thank you!