X | Y

Question

1 | 1
2 | 3
3 | 6
4 | 10
7 | 28
8 | 55
n | ?

Find expression that works for all steps using the variable x!?

Answer 1

Note that each time x increases by 1, y increases by x

I think there's a typo. When Y=55, x should be 9, not 8.

So, Y(x) = 1 + 2 + 3 + ... + x

Note that if you expand the terms a bit more,

1 + 2 + 3 + 4 + ... + x-3 + x-2 + x-1 + x

you can take them in pairs, working from both ends of the series. Each pair adds up to x+1

1+x + 2+x-1 + 3+x-2 + 4+x-3 ...

Since there are x terms, there are x/2 pairs, and the sum is thus

x(x+1)/2