Q: The spool for a cassette tape has the circumference of 70 mm. The thickness of the tape makes each turn is 0.075mm longer than the pervious one. How many times will the spool turn playing a tape that is 54 metres long?

I have attempted to answer it

using the sum formula

$\displaystyle S_{n} = \frac{n}{2}(2a+(n-1)d) $

a = 70 d = common difference = 0.075

$\displaystyle S_{n} = \frac{n}{2}(140+(n-1)0.075) = 54$

$\displaystyle 108000= 140000n + 75n^2-75n $

in the end I get n = -1866.4382or0.771523

which is clearly wrong

I cant seem to find the answer when I try to use the nth term

please help