induction thing

[manwonders]

I want to prove on the induction that there is room for 5000 balls in the field.
the field is filled, to hang around by first putting 100
balls there and by then adding balls which at time 10 balls more than previous.
how many times is it required that the field becomes full?

is it like this;
100+10+20+30+40+50+60+70+80+90+100+110... until there is 5000 balls??

I guess this is wrong?

 

[galactus]

I do not see why you would need induction. Induction is to show that if something works for the ith case, then it works for the (i+1)th case. You have a definitive amount of 5000.

It is \(\displaystyle 100+10\sum_{t=1}^{n}t=5000\)

\(\displaystyle 100+10\cdot\frac{n(n+1)}{2}=5000\)

Solving we find n is between 30 and 31.

n=30 gives 4750 and n=31 gives 5060

Precisely, \(\displaystyle n=\frac{\sqrt{3921}-1}{2}\approx 30.8089\)