So the arithmetic series is -4-3-2-1+0+1+2+... and the geometric series is .

The nth term in the arithmetic series is -4+ (n-1) and the sum up to that n is (-4+ -4+ n-1)n/2= n(n- 9)/2, exactly what you have.

The sum of the geometric series is which isalmostwhat you have. Looks like you multiplied by 1/2 rather than dividing.

Now you want

Notice that theinfiniteserie would have sum 16. n(n-9)/2> 16 when or .

when which is about 11.7.

Now just check: when n= 11, the arithmetic sum is 11(11-9)/2= 11 and the geometric sum is . When n= 12, the arithmetic sum is 12(12- 9)/2= 18 and the geometric sum is [tex]16(1- .5^{11})= 15.9921875.